**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:** 196002b:T43f,**The Setup:** This problem requires calculating the combined number of digits of two large numbers with a shared exponent. We can determine the number of digits by sandwiching the values between sequential powers of $10$. **Step 1:** Define the digit counting function. Let $d_1$ be the number of digits in $2^{2024}$. Thus, $10^{d_1-1} < 2^{2024} < 10^{d_1}$. Let $d_2$ be the number of digits in $5^{2024}$. Thus, $10^{d_2-1} < 5^{2024} < 10^{d_2}$. **Step 2:** Multiply the two bounding inequalities. $$10^{d_1-1} \times 10^{d_2-1} < 2^{2024} \times 5^{2024} < 10^{d_1} \times 10^{d_2}$$ $$10^{d_1+d_2-2} < (2 \times 5)^{2024} < 10^{d_1+d_2}$$ $$10^{d_1+d_2-2} < 10^{2024} < 10^{d_1+d_2}$$ **Step 3:** Evaluate the integer constraints on the exponents. For the exact integer $10^{2024}$ to be strictly bounded between these two powers of $10$, the exponent $2024$ must equal the lower bound exponent plus one: $$2024 = (d_1 + d_2 - 2) + 1$$ $$2024 = d_1 + d_2 - 1$$ $$d_1 + d_2 = 2025$$ The total number of consecutive digits written on the page is $2025$. **Final Answer:** 20252c:T642,**The Setup:** We are asked to find the tangent of the angle between two tangents drawn from an external point to a circle. We will use the geometric properties of right triangles formed by the tangents, the circle's radius, and the distance to the center. **Step 1:** Determine the circle's center and radius. The equation is $x^2 + y^2 - 14x + 16y + 88 = 0$. Complete the square for $x$ and $y$: $$(x^2 - 14x + 49) + (y^2 + 16y + 64) = -88 + 49 + 64$$ $$(x - 7)^2 + (y + 8)^2 = 25$$ Center $C = (7, -8)$ and Ra

IPMAT Indore 2024 Paper — Full Mock