Binomial Theorem
Coefficients in Ratio — Finding n and Specific Coefficient
nta_pyq_2023_apr
Grade 11
Question:
If the coefficients of three consecutive terms in the expansion of $(1+x)^n$ are in the ratio $1:5:20$, then the coefficient of the fourth term is
Step-by-Step Solution
Key Concept: $\frac{\binom{n}{r}}{\binom{n}{r-1}}=\frac{n-r+1}{r}=5$ and $\frac{\binom{n}{r+1}}{\binom{n}{r}}=\frac{n-r}{r+1}=4$.
$n=29,\ r=5$. $\binom{29}{3}=\frac{29\times28\times27}{6}=3654$.
Correct Answer: 4