Question

For an integer $n \geq 2$, if the arithmetic mean of all coefficients in the binomial expansion of $(x + y)^{2n-3}$ is 16, then the distance of the point $P(2n-1, n^2-4n)$ from the line $x + y = 8$ is:

• A. $\sqrt{2}$
• B. $2\sqrt{2}$
• C. $5\sqrt{2}$
• D. $3\sqrt{2}$

Hint:

The arithmetic mean of coefficients in a binomial expansion can be used to find the value of $n$.

Answer 1

The Correct Option is D

1. Determine the number of terms in $(x + y)^{2n-3}$: \[ \text{Number of terms} = 2n - 2 \]
2. Sum of all coefficients: \[ \text{Sum of coefficients} = 2^{2n-3} \]
3. Arithmetic mean of all coefficients: \[ \text{Arithmetic mean} = \frac{2^{2n-3}}{2n-2} = 16 \] \[ 2^{2n-3} = 16(2n-2) \] \[ 2^{2n-3} = 2^4(n-1) \] \[ 2n-3 = 4 \implies n = 5 \]
4. Determine the point $P$: \[ P(2n-1, n^2-4n) = P(9, 5) \] 5. Calculate the distance from the line $x + y = 8$: \[ \text{Distance} = \left| \frac{9 + 5 - 8}{\sqrt{2}} \right| = \frac{6}{\sqrt{2}} = 3\sqrt{2} \] Therefore, the correct answer is (4) $3\sqrt{2}$.