Question:

**Step 1:** Use the ratio of consecutive binomial coefficients. **Step 2:** The ratio of the first two given binomial coefficients is: $$ \frac{\binom{n}{r}}{\binom{n}{r-1}} = \frac{84}{36} = \frac{7}{3} $$ **Step 3:** Using the general formula for the ratio of consecutive binomial coefficients, $\frac{\binom{n}{k}}{\binom{n}{k-1}} = \frac{n - k + 1}{k}$, we set $k=r$: $$ \frac{n - r + 1}{r} = \frac{7}{3} $$ **Step 4:** Cross-multiplication yields $3(n - r + 1) = 7r$, which simplifies to $3n - 3r + 3 = 7r$. **Step 5:** This results in the linear equation: $$ 10r - 3n = 3 \quad \text{(Equation 1)} $$ **Step 6:** Similarly, consider the ratio of the second and third given binomial coefficients: $$ \frac{\binom{n}{r+1}}{\binom{n}{r}} = \frac{126}{84} = \frac{3}{2} $$ Using the general formula $\frac{\binom{n}{k+1}}{\binom{n}{k}} = \frac{n - k}{k + 1}$, we set $k=r$: $$ \frac{n - r}{r + 1} = \frac{3}{2} $$ **Step 7:** Cross-multiplication yields $2(n - r) = 3(r + 1)$, which simplifies to $2n - 2r = 3r + 3$. This results in the linear equation: $$ 2n - 5r = 3 \quad \text{(Equation 2)} $$ **Step 8:** Solve the system of linear equations (Equation 1) and (Equation 2): 1) $10r - 3n = 3$ 2) $2n - 5r = 3$ Multiply Equation (2) by 2: $2(2n - 5r) = 2(3)$ $4n - 10r = 6 \quad \text{(Equation 3)}$ Rearrange Equation (1) to align terms: $-3n + 10r = 3 \quad \text{(Equation 4)}$ Add Equation (3) and Equation (4): $(4n - 10r) + (-3n + 10r) = 6 + 3$ $n = 9$ Substitute $n=9$ into Equation (2): $2(9) - 5r = 3$ $18 - 5r = 3$ $15 = 5r$ $r = 3$ Thus, the value of $r$ is 3.