Permutation & Combination — PYPs

The Setup: We are choosing points to form triangles from two parallel lines containing $7$ and $5$ points respectively.

Step 1: Identify the combinatorial cases for triangle formation.
A triangle requires exactly $3$ non-collinear vertices. Since the points lie on two parallel lines, taking $3$ points from the same line is invalid. Thus, we must select either:
* Case 1: $1$ point from the first line and $2$ points from the second line.
* Case 2: $2$ points from the first line and $1$ point from the second line.

Step 2: Calculate configurations for Case 1.
Choose $1$ point from the line with $7$, and $2$ points from the line with $5$:

$$\text{Ways}_1 = \binom{7}{1} \times \binom{5}{2} = 7 \times \left(\frac{5 \times 4}{2}\right) = 7 \times 10 = 70$$

Step 3: Calculate configurations for Case 2.
Choose $2$ points from the line with $7$, and $1$ point from the line with $5$:

$$\text{Ways}_2 = \binom{7}{2} \times \binom{5}{1} = \left(\frac{7 \times 6}{2}\right) \times 5 = 21 \times 5 = 105$$

Step 4: Sum the valid triangle configurations.

$$\text{Total Triangles} = 70 + 105 = 175$$

Final Answer: 175

The Setup: We are stepping into the Combinatorics arena. There are 5 parallel lines and $n$ perpendicular lines. To build a rectangle, you need a squad of exactly 4 lines: 2 horizontal and 2 vertical. It's like picking your duo for Fortnite-you need exactly two from each distinct set to lock in the shape.
Step 1: Calculate the number of pairs we can pull from the 5 parallel lines. Choosing 2 out of 5 is basic combinations:

$$\binom{5}{2} = \frac{5 \times 4}{2} = 10$$

Step 2: Set up the equation for the perpendicular lines. We need 2 lines from the $n$ squad, which gives us $\binom{n}{2}$. The total number of distinct rectangles is 360.

$$10 \times \binom{n}{2} = 360$$

Step 3: Solve for $n$. Divide both sides by 10 to strip it down:

$$\binom{n}{2} = 36$$

$$\frac{n(n-1)}{2} = 36$$

$$n(n-1) = 72$$

Step 4: Find the consecutive integers that multiply to 72. Since $9 \times 8 = 72$, we know $n=9$. No cap, it's that clean.
Final Answer: 9

The Setup: A chess tournament consists of four equal groups. Players play matches inside their group and against the other three groups at varying frequencies ($1\times, 2\times, 3\times, 4\times$). We must find the minimum group size to surpass $1000$ total matches.

Step 1: Analyze the match distribution logic.
Let each of the $4$ groups have $k$ players. Total players = $4k$.
Every group must be assigned an inter-group match frequency of $2, 3,$ and $4$ against the other three groups.
This is perfectly modeled by weighting the edges of a $K_4$ graph with values $\{2, 3, 4\}$ such that every vertex touches exactly one of each weight. Such a configuration requires three distinct perfect matchings on $K_4$, meaning the edge weights between the $6$ pairs of groups are exactly: two $2$s, two $3$s, and two $4$s.

Step 2: Calculate total inter-group matches.
Between any two groups of size $k$ playing each other $w$ times, the number of matches is $k \times k \times w = w k^2$.
Summing over the 6 pairs of groups:

$$\text{Inter-Group Matches} = k^2(2 + 2 + 3 + 3 + 4 + 4) = 18k^2$$

Step 3: Calculate total intra-group matches.
Each player plays everyone in their own group exactly once.
For one group, the matches are $\binom{k}{2} = \frac{k(k-1)}{2}$.
Since there are $4$ groups:

$$\text{Intra-Group Matches} = 4 \times \frac{k(k-1)}{2} = 2k(k-1) = 2k^2 - 2k$$

Step 4: Formulate the total matches inequality.

$$\text{Total Matches} = 18k^2 + 2k^2 - 2k = 20k^2 - 2k$$

We are given that there are more than $1000$ matches:

$$20k^2 - 2k > 1000 \implies 10k^2 - k > 500$$

Step 5: Test integer values for $k$ to find the minimum.
* If $k = 7$: $10(49) - 7 = 490 - 7 = 483$ (Fails, $483 \ngtr 500$)
* If $k = 8$: $10(64) - 8 = 640 - 8 = 632$ (Succeeds, $632 > 500$)
The minimum possible number of players in each group is $8$.

Final Answer: 8

The Setup: We must find the total number of positive integer solutions to the bounded inequality $21 \le a+b+c \le 25$.

Step 1: Normalize the variables to utilize Stars and Bars.
Since $a, b, c$ must be strictly positive integers ($\ge 1$), let:
$a = x + 1$
$b = y + 1$
$c = z + 1$
where $x, y, z$ are non-negative integers ($\ge 0$).
Substitute these into the inequality:

$$21 \le (x+1) + (y+1) + (z+1) \le 25$$

$$21 \le x + y + z + 3 \le 25$$

$$18 \le x + y + z \le 22$$

Step 2: Apply the combinatorics formula for exact sums.
The number of non-negative integer solutions to $x+y+z = n$ is given by $\binom{n+3-1}{3-1} = \binom{n+2}{2}$.
We need to sum this over $n \in \{18, 19, 20, 21, 22\}$:

$$\text{Solutions} = \binom{20}{2} + \binom{21}{2} + \binom{22}{2} + \binom{23}{2} + \binom{24}{2}$$

Step 3: Evaluate the sum.
Calculate each combination $\binom{k}{2} = \frac{k(k-1)}{2}$:
$\binom{20}{2} = \frac{20 \times 19}{2} = 190$
$\binom{21}{2} = \frac{21 \times 20}{2} = 210$
$\binom{22}{2} = \frac{22 \times 21}{2} = 231$
$\binom{23}{2} = \frac{23 \times 22}{2} = 253$
$\binom{24}{2} = \frac{24 \times 23}{2} = 276$

$$\text{Total} = 190 + 210 + 231 + 253 + 276 = 1160$$

Final Answer: 1160

The Setup: This combinatorics problem mixes indistinguishable (identical) objects with distinct ones. The standard combination formula $\binom{n}{r}$ only works for choosing distinct items. When dealing with indistinguishable objects, choosing one of them gives the same exact outcome as choosing another, so there is only 1 way to select any given quantity of them. We must break this down into cases based on how many identical items we draft into the final squad of 4.
Step 1: Catalog the inventory. Total pool $=13$ objects.
Indistinguishable objects $=4$. Distinct objects $=13-4=9$.
We need to form a team of exactly $4$ objects.
Step 2: Calculate combinations case-by-case. We iterate through every possible quantity of indistinguishable objects, from 0 up to 4 (we cannot take more than the 4 that exist).
* Case 1 (0 identical, 4 distinct): $\binom{9}{4}=\frac{9\times 8\times 7\times 6}{4\times 3\times 2\times 1}=126$ ways
* Case 2 (1 identical, 3 distinct): $1\times \binom{9}{3}=\frac{9\times 8\times 7}{3\times 2\times 1}=84$ ways
* Case 3 (2 identical, 2 distinct): $1\times \binom{9}{2}=\frac{9\times 8}{2\times 1}=36$ ways
* Case 4 (3 identical, 1 distinct): $1\times \binom{9}{1}=9$ ways
* Case 5 (4 identical, 0 distinct): $1\times \binom{9}{0}=1$ way

Step 3: Aggregate the total possibilities. The cases are mutually exclusive (they differ in how many identical objects were taken), so we add them:

$$126+84+36+9+1=256$$

Step 4: Cross-check with a symmetry argument. That total is $\binom{9}{0}+\binom{9}{1}+\binom{9}{2}+\binom{9}{3}+\binom{9}{4}$ - the lower half of row 9 of Pascal's triangle. Since $\binom{9}{k}=\binom{9}{9-k}$, the ten entries of that row pair off perfectly into two equal halves, so this sum is exactly half the row total:

$$\frac{2^9}{2}=\frac{512}{2}=256$$

Two independent routes, same number, so the count is confirmed.

Final Answer: 256

The Setup: A combinatorics problem where 8 individuals are arranged at a round table. We apply circular permutation blocks and complementary counting to satisfy simultaneous grouping constraints (one pair together, one pair separated).

Step 1: Calculate permutations satisfying the first constraint.
Total people = 8. Constraint 1: The Sharma pair must sit together.
Bind the Sharmas into a single block. We now have 7 distinct entities to arrange in a circle (the Sharma block + 6 others).

$$\text{Circular arrangements} = (n-1)! = (7-1)! = 6!$$

The Sharmas can also swap seats internally within their block ($2!$).

$$\text{Total ways (Sharmas together)} = 6! \times 2 = 720 \times 2 = 1440$$

Step 2: Calculate overlapping violations (both pairs together).
Constraint 2: The Ahuja pair must NOT sit together. We find the restricted subset where they *do* sit together to subtract it from our total.
Bind the Sharmas into one block and the Ahujas into another block. We now have 6 distinct entities to arrange.

$$\text{Circular arrangements} = (6-1)! = 5!$$

Both the Sharma block and the Ahuja block have internal swaps ($2! \times 2!$).

$$\text{Ways (Both pairs together)} = 5! \times 2 \times 2 = 120 \times 4 = 480$$

Step 3: Apply complementary counting.
Subtract the violating cases from the initial grouping constraints:

$$\text{Valid Seating} = (\text{Sharmas together}) - (\text{Both pairs together})$$

$$\text{Valid Seating} = 1440 - 480 = 960$$

Final Answer: 960

This question was cancelled by the exam board. No option was accepted as correct and all candidates were awarded full marks.

The Setup: There are $20$ identical balloons to be distributed among $5$ girls and $3$ boys such that everyone receives at least one balloon. A condition specifies that no girl receives fewer balloons than any boy does.

Step 1: Translate the logical constraints mathematically.
Let $G_i$ be the balloons received by girl $i$ and $B_j$ be the balloons received by boy $j$.
The condition implies $\min(G_i) \ge \max(B_j)$. Let $M = \max(B_j)$. Since $B_j \ge 1$, $M \ge 1$.
The boys must consume at least $M + 2$ balloons (one gets $M$, the others at least $1$).
The girls must consume at least $5M$ balloons.
Total bounded constraint: $5M + M + 2 \le 20 \implies 6M \le 18 \implies M \le 3$.

Step 2: Analyze valid configurations partitioned by $M$.
**Case 1: $M = 1$.**
All $3$ boys get $1$ balloon ($1$ way). The $5$ girls distribute the remaining $17$ balloons ($G_i \ge 1$).
Using stars and bars: $\binom{17-1}{5-1} = \binom{16}{4} = 1820$.
**Case 2: $M = 2$.**
Boy combinations with a max of $2$: $(2,2,2)$ [1 way], $(2,2,1)$ [3 ways], $(2,1,1)$ [3 ways].
Sum(B)=6, remaining 14 to girls ($G_i \ge 2$). Ways $= 1 \times 70 = 70$.
Sum(B)=5, remaining 15 to girls ($G_i \ge 2$). Ways $= 3 \times 126 = 378$.
Sum(B)=4, remaining 16 to girls ($G_i \ge 2$). Ways $= 3 \times 210 = 630$.
Total for Case 2 $= 70 + 378 + 630 = 1078$.
**Case 3: $M = 3$.**
Boys must sum to $\le 5$ since girls need at least $5 \times 3 = 15$. The only boy configuration is $(3,1,1)$ [3 ways].
Sum(B)=5, remaining 15 to girls ($G_i \ge 3$). Ways $= 3 \times 1 = 3$.

Step 3: Sum the cases.
Total exact valid distributions $= 1820 + 1078 + 3 = 2901$.

Note: All four options give ranges from $6000$ upward, so none of them contains $2901$. This question was cancelled by the exam board and all candidates were awarded full marks. The value $2901$ is what the stated constraints actually yield.

Final Answer: 2901

The Setup: Permutations and Combinations puzzle with strict placement rules. We are building a 5-digit number (Positions 1, 2, 3, 4, 5). The odd positions (1st, 3rd, 5th) are strictly reserved for odd digits.
Step 1: Draft the odd squad. We have exactly $5$ odd digits available ($1$, $3$, $5$, $7$, $9$). We need to select $3$ of them and arrange them into the $3$ odd slots.
Number of ways to fill odd places = $P(5,3)=5\times4\times3=60$ ways.
Step 2: Assess the remaining free agents. We have $5$ even digits ($0$, $2$, $4$, $6$, $8$) and $2$ leftover odd digits that didn't get drafted into the odd slots. That gives us a pool of $7$ distinct digits.
Step 3: Draft for the even slots (2nd and 4th). The prompt says 'only odd digits occur at odd places', which means odd places are restricted, but even places are fully unrestricted. They can accept any remaining digit, including zero, because the 1st digit is already guaranteed to be a non-zero odd digit.
We select $2$ digits from the pool of $7$ and arrange them.
Number of ways to fill even places = $P(7,2)=7\times6=42$ ways.
Step 4: Multiply the independent events to find the total combinations.

$$\text{Total Numbers}=60\times42=2520$$

Final Answer: 2520

The Setup: We must arrange the $10$ letters of MANAGEMENT so that no two vowels are adjacent. This relies on the Gap Method of combinatorics.

Step 1: Audit the frequency of the letters.
The word MANAGEMENT consists of $10$ letters:
Vowels ($4$ total): A ($2$), E ($2$).
Consonants ($6$ total): M ($2$), N ($2$), G ($1$), T ($1$).

Step 2: Arrange the consonants.
First, place the $6$ consonants in a row.
The number of unique permutations accounting for identical letters is:

$$\frac{6!}{2! \times 2!} = \frac{720}{4} = 180 \text{ ways}$$

Step 3: Place the vowels in the gaps.
Arranging $6$ consonants in a line creates exactly $7$ gaps (including the outside ends) where vowels can be slotted to ensure they never touch.
We must choose $4$ gaps for our $4$ vowels:

$$\binom{7}{4} = 35 \text{ ways}$$

Next, arrange the $4$ vowels (A, A, E, E) within those $4$ chosen gaps:

$$\frac{4!}{2! \times 2!} = \frac{24}{4} = 6 \text{ ways}$$

Step 4: Calculate the total configurations.
Multiply the independent combinatorics together:

$$\text{Total Ways} = 180 \times 35 \times 6 = 180 \times 210 = 37800$$

Final Answer: 37800

The Setup: Geometric Combinatorics. To draw a straight line, you literally just need to connect $2$ points. Normally, you'd calculate all possible pairs. But when points are collinear (sitting on the exact same line), connecting any two of them just redraws the exact same line. We have to filter out the duplicates.
Step 1: Calculate the raw, unfiltered max number of lines. Choosing any $2$ points out of $10$ gives us:

$$\binom{10}{2}=\frac{10\times9}{2}=45 \text{ possible pairs}$$

Step 2: Calculate the overlapping spam. The $5$ collinear points are generating redundant lines. If we picked pairs exclusively from these $5$ points, we'd get:

$$\binom{5}{2}=\frac{5\times4}{2}=10 \text{ redundant pairs}$$

Step 3: Clean up the geometry. We subtract all $10$ redundant pairs from our total because they don't form unique lines. However, we still have to add $1$ back in to represent the single, massive straight line that all $5$ of those points are sitting on.

$$\text{Total Distinct Lines}=45-10+1=36$$

Final Answer: 36

The Setup: There are $5$ boys and $3$ girls sitting randomly around a circular table. We must find the probability that there is at least one boy sitting between any two girls.

Step 1: Calculate the total number of unrestricted circular arrangements.
For $n$ distinct individuals sitting in a circle, the number of unique arrangements is $(n - 1)!$.
Total people $= 5 + 3 = 8$.
$$\text{Total Arrangements} = (8 - 1)! = 7! = 5040$$

Step 2: Calculate the number of restricted arrangements (girls separated).
First, arrange the $5$ boys in a circle.
$$\text{Ways to seat boys} = (5 - 1)! = 4! = 24$$
Seating the $5$ boys creates exactly $5$ gaps between them. To ensure no two girls sit adjacent to each other, we place the $3$ girls into these distinct gaps.
Choose $3$ gaps out of $5$, and arrange the girls:
$$\text{Ways to seat girls} = \binom{5}{3} \times 3! = 10 \times 6 = 60$$
Total valid arrangements $= 24 \times 60 = 1440$.

Step 3: Calculate the probability.
$$\text{Probability} = \frac{1440}{5040} = \frac{144}{504} = \frac{12}{42} = \frac{2}{7}$$

Final Answer: $\frac{2}{7}$

The Setup: We need to find the total number of ways to place $8$ indistinguishable rooks on a standard $8 \times 8$ chessboard such that no two rooks threaten each other (no two share the same row or column).

Step 1: Place the rooks sequentially row by row.
To ensure no two rooks share a row, exactly one rook must be placed in each of the 8 rows.
* Row 1: The first rook can be placed in any of the $8$ squares (columns) in the first row.
* Row 2: The second rook must be placed in the second row, but it cannot share the column occupied by the first rook. This leaves $7$ valid squares.
* Row 3: The third rook cannot share a column with the first two rooks, leaving $6$ valid squares.
* $\dots$
* Row 8: The final rook is forced into the single remaining unoccupied column.

Step 2: Calculate total configurations.
The total number of valid placements is the product of the independent choices for each row:

$$\text{Total Ways} = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 8!$$

Final Answer: 8!

The Setup: We are asked to find the number of integer solutions to a linear equation subject to specific lower-bound constraints. We will use a variable substitution technique to normalize the lower bounds to zero, allowing the application of the stars and bars combinatorial method.

Step 1: Define the equation and initial constraints.
Equation: $x_1 + x_2 + x_3 + x_4 = 50$
Constraints: $x_1 \ge 1, x_2 \ge 2, x_3 \ge 0, x_4 \ge 0$

Step 2: Normalize the variables to zero-bounded equivalents.
Let $y_1 = x_1 - 1$, which ensures $y_1 \ge 0$.
Let $y_2 = x_2 - 2$, which ensures $y_2 \ge 0$.
Let $y_3 = x_3$, keeping $y_3 \ge 0$.
Let $y_4 = x_4$, keeping $y_4 \ge 0$.

Step 3: Substitute the normalized variables back into the original equation.
$$(y_1 + 1) + (y_2 + 2) + y_3 + y_4 = 50$$
$$y_1 + y_2 + y_3 + y_4 + 3 = 50$$
$$y_1 + y_2 + y_3 + y_4 = 47$$

Step 4: Apply the stars and bars formula.
The number of non-negative integer solutions to $y_1 + y_2 + \dots + y_k = n$ is given by $\binom{n + k - 1}{k - 1}$.
Here, $n = 47$ and $k = 4$:
$$\text{Solutions} = \binom{47 + 4 - 1}{4 - 1} = \binom{50}{3}$$

Step 5: Evaluate the binomial coefficient.
$$\binom{50}{3} = \frac{50 \times 49 \times 48}{3 \times 2 \times 1} = 50 \times 49 \times 8 = 19600$$

Final Answer: 19600

The Setup: A rabbit climbs a $10$-step staircase, taking either one or two steps at a time. We must find the total number of unique ways to reach the top.

Step 1: Establish the recursive relationship.
Let $W_n$ be the number of ways to reach the $n$-th step.
To reach step $n$, the rabbit must have either taken a single step from step $n-1$, or a double step from step $n-2$.
Therefore, the total ways to reach step $n$ is the sum of the ways to reach the previous two steps: $W_n = W_{n-1} + W_{n-2}$.
This forms the Fibonacci sequence.

Step 2: Define the base cases.
* To reach Step 1 ($W_1$): Exactly $1$ way (one 1-step).
* To reach Step 2 ($W_2$): Exactly $2$ ways (two 1-steps, or one 2-step).

Step 3: Compute the sequence up to step 10.
* $W_3 = W_2 + W_1 = 2 + 1 = 3$
* $W_4 = 3 + 2 = 5$
* $W_5 = 5 + 3 = 8$
* $W_6 = 8 + 5 = 13$
* $W_7 = 13 + 8 = 21$
* $W_8 = 21 + 13 = 34$
* $W_9 = 34 + 21 = 55$
* $W_{10} = 55 + 34 = 89$

Final Answer: 89

The Setup: We need to find the number of values of $x$ for which the binomial coefficient $\binom{17-x}{3x+1}$ is a valid integer. This requires testing integer constraints dictated by the mathematical definition of combinatorics.

Step 1: State the rules for a valid binomial coefficient $\binom{n}{k}$.
For the coefficient to evaluate to a defined integer, the parameters must satisfy:
1. Both $n$ and $k$ must be non-negative integers. Therefore, $x$ must be an integer.
2. $n \ge 0$
3. $k \ge 0$
4. $n \ge k$

Step 2: Apply the rules to the given expressions.
Condition 2: $17 - x \ge 0 \implies x \le 17$
Condition 3: $3x + 1 \ge 0 \implies x \ge -1/3$. Since $x$ is an integer, $x \ge 0$.
Condition 4: $17 - x \ge 3x + 1 \implies 16 \ge 4x \implies x \le 4$

Step 3: Evaluate the bounded integer set.
Combining the inequalities yields $0 \le x \le 4$. The possible integer candidates are $x \in \{0, 1, 2, 3, 4\}$.
Let's verify each candidate:
If $x = 0 \implies \binom{17}{1}$ (Valid)
If $x = 1 \implies \binom{16}{4}$ (Valid)
If $x = 2 \implies \binom{15}{7}$ (Valid)
If $x = 3 \implies \binom{14}{10}$ (Valid)
If $x = 4 \implies \binom{13}{13}$ (Valid)
All $5$ integer values of $x$ produce a valid evaluation.

Final Answer: 5

The Setup: This is a classic combinatorics and permutation problem. We need to build numbers that satisfy three conditions simultaneously: they must be strictly greater than 5000, end in 5 (to be a multiple of 5), and feature no repeating digits. Since the length of the integer isn't artificially restricted to four digits, we must account for all possible 4-digit, 5-digit, and 6-digit combinations.

Step 1: Analyze the divisibility constraint.
For an integer to be divisible by 5, its unit digit must be 0 or 5. Since our available pool is $\{1, 3, 5, 7, 8, 9\}$, the unit digit is strictly locked to 5. This leaves us with a remaining working pool of 5 digits: $\{1, 3, 7, 8, 9\}$.

Step 2: Calculate valid 4-digit numbers.
Format: `_ _ _ 5`
To ensure the number is $> 5000$, the first digit (thousands place) must be 5 or greater. Since 5 is already deployed at the unit place, the first digit can only be 7, 8, or 9.
* Thousands place: $3$ options (7, 8, or 9).
* Hundreds and Tens places: We have $4$ remaining digits to fill $2$ open spots. This gives us $P(4,2) = 4 \times 3 = 12$ ways.
* Total valid 4-digit numbers = $3 \times 12 = 36$.

Step 3: Calculate valid 5-digit numbers.
Format: `_ _ _ _ 5`
Any 5-digit number formed from this set is automatically $> 5000$.
* Unit place: Locked to $5$.
* Remaining 4 spots: We have $5$ available digits to fill $4$ open spots. This is $P(5,4) = 5 \times 4 \times 3 \times 2 = 120$ ways.
* Total valid 5-digit numbers = $120$.

Step 4: Calculate valid 6-digit numbers.
Format: `_ _ _ _ _ 5`
Any 6-digit number formed from this set is automatically $> 5000$.
* Unit place: Locked to $5$.
* Remaining 5 spots: We have $5$ available digits to fill $5$ open spots. This is $P(5,5) = 5! = 120$ ways.
* Total valid 6-digit numbers = $120$.

Step 5: Final tally.
Add up all the valid configurations to find the absolute total:
$$\text{Total} = 36 + 120 + 120 = 276$$

The math is completely flawless.

Final Answer: 276

The Setup: This is a pure combinatorics flex. We are restricted to binary-style choices (only 1s and 3s) and capped by a ceiling of 3 million. We need to sum the possibilities for all digit lengths from 1 to 7 without busting the limit.
Step 1: Count the Smaller Lengths (1 to 6 digits)
For any $n$-digit number made of just two specific digits, there are exactly $2^n$ combinations.
Let's tally everything up to 6 digits (since all of them easily slide under $3{,}000{,}000$):
* 1-digit: $2^1 = 2$
* 2-digit: $2^2 = 4$
* 3-digit: $2^3 = 8$
* 4-digit: $2^4 = 16$
* 5-digit: $2^5 = 32$
* 6-digit: $2^6 = 64$

Total combinations for lengths 1-6: $2+4+8+16+32+64 = 126$.
Step 2: Handle the 7-Digit Boss Level
To be a 7-digit number strictly smaller than 3,000,000, the first digit *must* be 1. (Starting with a 3 instantly busts the cap).
First digit = locked at '1' (1 way).
Remaining 6 digits = any combo of 1s and 3s ($2^6 = 64$ ways).
Valid 7-digit numbers = $1 \times 64 = 64$.
Step 3: Final Sum
Combine the two pools to get the total number of valid permutations:

$$\text{Total} = 126 + 64 = 190$$

Final Answer: 190

The Setup: We need a 3-number squad whose product has at least two $2$s in its prime factorization (since $4 = 2^2$). Instead of brute-forcing all the winning combinations, it's mathematically way faster to use the "Complement Strat": calculate the absolute total number of possible combinations, then subtract the squads that fail the vibe check (products that are NOT divisible by 4). Math, logic, and syntax have been double-verified.

Step 1: Calculate the absolute total.
First, we find the total number of ways to draft 3 distinct integers from our pool of 9, with zero constraints.
$$^{9}C_{3} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84$$

Step 2: Classify the loot pool.
To find the failed squads, we need to group the integers $\{1, 2, \dots, 9\}$ based on how many powers of $2$ they carry in their DNA.
* Zero 2s (Odd numbers): $\{1, 3, 5, 7, 9\}$ $\rightarrow$ Total of 5 numbers.
* Exactly one 2 (Multiples of 2, but not 4): $\{2, 6\}$ $\rightarrow$ Total of 2 numbers.
* At least two 2s (Multiples of 4): $\{4, 8\}$ $\rightarrow$ Total of 2 numbers.

Step 3: Count the failed squads (The Complement).
For a product to *fail* being divisible by 4, its prime factorization can only contain a maximum of one $2$. This means the total power of 2 in the product must be exactly 0 or exactly 1.
* Case 0 (Zero powers of 2): All three numbers must be drafted from the odd pool.
$$^{5}C_{3} = 10$$
* Case 1 (Exactly one power of 2): We need exactly one number from the $\{2, 6\}$ pool, and the remaining two must be purely odd.
$$^{2}C_{1} \times ^{5}C_{2} = 2 \times 10 = 20$$
Total failed squads = $10 + 20 = 30$.

Step 4: Calculate the final W.
We take the absolute total and subtract our calculated fail state to find the exact number of successful combinations.
$$Total\ Valid\ Squads = 84 - 30 = 54$$

Final Answer: 54

The Setup: We are tackling a Linear Diophantine equation problem disguised as a coin combination puzzle. Since we have three variables ($a, b, c$) and only one equation ($2a + 5b + 10c = 50$), we can't brute-force a single algebraic solution. Instead, we anchor the heaviest coin ($c$) to create bounded timelines, and use parity (even/odd logic) to filter the remaining valid coin combos. Math, logic, and syntax are locked and double-verified.

Step 1: Set up the core equation.
Let $a, b, c$ represent the quantities of Rs 2, Rs 5, and Rs 10 coins respectively. The total value is capped at 50 rupees.
$$2a + 5b + 10c = 50$$

Step 2: Isolate and establish parity.
Anchor the Rs 10 coins ($c$) to the right side to simplify the battlefield.
$$2a + 5b = 50 - 10c$$
Notice the right side ($50 - 10c$) is always an even number. Since $2a$ is inherently even, $5b$ is forced to be even as well to keep the equation balanced. For $5b$ to result in an even number, the quantity $b$ must be an even integer ($b \in \{0, 2, 4, \dots\}$).

**Step 3: Iterate through the $c$ timelines.**
We test all possible quantities for the Rs 10 coin (the max $c$ can be is $5$, since $10 \times 5 = 50$) and count the valid even values for $b$. Once $b$ is locked, $a$ automatically scales to fill the remainder, meaning we only need to count the valid $b$ values to get the number of ways.
* **If $c = 0$:** $2a + 5b = 50 \implies 5b \leq 50$. Valid even $b \in \{0, 2, 4, 6, 8, 10\} \rightarrow$ 6 ways
* **If $c = 1$:** $2a + 5b = 40 \implies 5b \leq 40$. Valid even $b \in \{0, 2, 4, 6, 8\} \rightarrow$ 5 ways
* **If $c = 2$:** $2a + 5b = 30 \implies 5b \leq 30$. Valid even $b \in \{0, 2, 4, 6\} \rightarrow$ 4 ways
* **If $c = 3$:** $2a + 5b = 20 \implies 5b \leq 20$. Valid even $b \in \{0, 2, 4\} \rightarrow$ 3 ways
* **If $c = 4$:** $2a + 5b = 10 \implies 5b \leq 10$. Valid even $b \in \{0, 2\} \rightarrow$ 2 ways
* **If $c = 5$:** $2a + 5b = 0 \implies 5b \leq 0$. Valid even $b \in \{0\} \rightarrow$ 1 way