For some $n \neq 10$, let the coefficients of the 5th, 6th and 7th terms in the binomial expansion of $(1 + x)^{n+4}$ be in A.P. Then the largest coefficient in the expansion of $(1 + x)^{n+4}$ is:
Step-by-Step Solution
Key Concept: If ${}^{n+4}C_4, {}^{n+4}C_5, {}^{n+4}C_6$ are in A.P., then $2\cdot{}^{n+4}C_5 = {}^{n+4}C_4 + {}^{n+4}C_6$. Solve for $n$ and find the largest binomial coefficient.
Setting up the A.P. condition: $4 \times {}^{n+4}C_5 = {}^{n+5}C_5 + {}^{n+5}C_6$, which simplifies to $n^2 - 13n + 30 = 0$, giving $n = 3$ or $n = 10$. Since $n \neq 10$, $n = 3$, so the expansion is $(1+x)^7$. The largest coefficient is the middle term $= {}^7C_4 = {}^7C_3 = 35$.
Correct Answer: 35