JEE PYQ: Permutation And Combination Question 30
Question 30 - 2020 (09 Jan Shift 2)
If $C_r = {}^{25}C_r$ and $C_0 + 5 \cdot C_1 + 9 \cdot C_2 + \cdots + (101) \cdot C_{25} = 2^{25} \cdot k$, then $k$ is equal to ______.
Show Answer
Answer: 51
Solution
$\sum_{r=0}^{25}(4r+1){}^{25}C_r = 4\sum r \cdot {}^{25}C_r + \sum {}^{25}C_r = 4 \times 25 \cdot 2^{24} + 2^{25} = 100 \cdot 2^{24} + 2^{25} = 2^{25}(50+1) = 51 \cdot 2^{25}$. So $k = 51$.
BETA
