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If C_r= _{}^{25}\textrm{C}_r and C_0+5\cdot C_1+9\cdot C_2+\cdots \cdots +(101)\cdot C_{25}=2^{25}\cdot k, then k is equal to _________.

 

Option: 1

15


Option: 2

30


Option: 3

51


Option: 4

27


Answers (1)

best_answer

Properties of Binomial Theorem and Binomial Coefficient

\sum_{r=1}^{n} r \times ^{n} C_{r} = n.2^{n-1}

 

Now,

General term for given series is: (4r + 1).25Cr

Applying summation

\\=4 \sum_{r=1}^{25} r \times \;\; ^{25} C_{r}+2^{25}\\ =4\times25\cdot 2^{24}+2^{25}=50\cdot2^{25}+2^{25}=(50+1)2^{25}=51\cdot2^{25}

k = 51

 

Posted by

seema garhwal

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