Binomial Theorem
Mean of Coefficients of Powers in Binomial Expansion
nta_pyq_2023_apr
Grade 11
Question:
The mean of the coefficients of $x,x^2,\ldots,x^7$ in the binomial expression of $(2+x)^9$ is _________.
Step-by-Step Solution
Key Concept: Coefficients of $x^k$ in $(2+x)^9$ are $\binom{9}{k}2^{9-k}$. Sum for $k=1$ to $7$ $=3^9-\binom{9}{0}2^9-\binom{9}{8}2^1-\binom{9}{9}$.
Sum of coefficients of $x$ through $x^7$ $=3^9-512-18-1=19152$. Mean $=2736$.
Correct Answer: 2736