Gujarati
7.Binomial Theorem
normal

Let $\left(\frac{n}{k}\right)=\frac{n !}{k !(n-k) !}$. Then the sum $\frac{1}{2^{10}} \sum \limits_{ k =0}^{10}\left(\frac{10}{ k }\right) k ^2$, lies in the interval

A

$(26,27)$

B

$(27,28)$

C

$(28,29)$

D

$(29,30)$

(KVPY-2021)

Solution

(b)

$\sum \limits_{ k =0}^{10}{ }^{10} C _{ k } \cdot k ^2=\sum \limits_{ k =1}^{10} \frac{10}{ k } \cdot{ }^9 C _{ k -1} \cdot k ^2$

$10 \sum\limits_{ k =1}^{10}( k -1+1)^9 C _{ k -1}=10\left(\sum \limits_{ k =1}^{10} 9 \cdot{ }^8 C _{ k -2}+\sum \limits_{ k =1}^{10}{ }^9 C _{ k -1}\right)$

$10\left(9 \cdot 2^8+2^9\right)=10 \cdot 2^8(9+2)=110 \cdot 2^8$

$\frac{1}{2^{10}} \times 28 \times 110=\frac{2^9}{2^{10}} \cdot(55)=\frac{55}{2}$

Standard 11
Mathematics

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