JEE Mains · Maths · STD 11 - 7. binomial theoram
If \({ }^{20} \mathrm{C}_{\mathrm{r}}\) is the co-efficient of \(\mathrm{x}^{\mathrm{r}}\) in the expansion of \((1+x)^{20}\), then the value of \(\sum_{r=0}^{20} r^{2}\,\,{ }^{20} C_{r}\) is equal to :
- A \(420 \times 2^{19}\)
- B \(380 \times 2^{19}\)
- C \(380 \times 2^{18}\)
- D \(420 \times 2^{18}\)
Answer & Solution
Correct Answer
(D) \(420 \times 2^{18}\)
Step-by-step Solution
Detailed explanation
\(\sum_{r=0}^{20} r^{2} \cdot{ }^{20} C_{r}\) \(\sum(4(r-1)+r) \cdot{ }^{20} C_{r}\) \(\sum r(r-1) \cdot \frac{20 \times 19}{r(r-1)} \cdot{ }^{18} C_{r}+r \cdot \frac{20}{r} \cdot \sum^{19} C_{r-1}\) \(\Rightarrow 20 \times 19.2^{18}+20.2^{19}\) \(\Rightarrow 420 \times 2^{18}\)
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