Binomial Theorem
Binomial Theorem
nta_abhyas_2025
Grade 11
Question:
If $C_0, C_1, C_2, \ldots$ are the binomial coefficients where $C_r = {}^nC_r$. Let $S = C_0 + C_1 + 2C_2 + 3C_3 + \cdots + nC_n$, then $S$ is equal to
Step-by-Step Solution
Key Concept: Factor out $n$ from $r \cdot ^nC_r$ to convert it to $^{n-1}C_{r-1}$, then sum using the binomial theorem.
We compute $S = \sum_{r=1}^{n} r \cdot ^nC_r$. Using the identity $r \cdot ^nC_r = r \cdot \frac{n!}{r!(n-r)!} = n \cdot \frac{(n-1)!}{(r-1)!(n-r)!} = n \cdot ^{n-1}C_{r-1}$, we have $S = n \sum_{r=1}^{n} ^{n-1}C_{r-1} = n \sum_{s=0}^{n-1} ^{n-1}C_s = n \cdot 2^{n-1}$. This uses the substitution $s = r-1$ and the binomial sum $\sum_{s=0}^{n-1} ^{n-1}C_s = 2^{n-1}$.
Correct Answer: n·2^(n-1)