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Q.

Let $\sum_{i = 0}^{100} \sum_{j = 0}^{100} (C_{i}^{2} + C_{j}^{2} + C_{i} C_{j}) = a .^{2 n} C_{n} + 2^{b}$ where a, b, n are even natural numbers and C_r represent the coefficient of x^r in the expansion of $(1 + x)^{100}$ then which is/are CORRECT ?

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a

a + 2b + 3n = 900

b

a – n = 102

c

$a + b + n = 504$

d

$a + b = 402$

answer is B, C.

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Detailed Solution

$\begin{array}{l} \sum_{i = 0}^{100100} \sum_{j = 0} (C_{i}^{2} + C_{j}^{2} + C_{i} C_{j}) = a^{2 n} C_{n} + 2^{b} \\ \sum_{i = 0}^{100} (101 C_{i}^{2} +^{200} C_{100} + C_{i} 2^{100}) \\ 101^{200} C_{100} + 101^{200} C_{100} + 2^{100} \cdot 2^{100} \\ 202^{200} C_{100} + 2^{200} = a^{2 n} C_{n} + 2^{b} \\ a = 202; n = 100; b = 200 \end{array}$

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Let ∑i=0100 ∑j=0100 Ci2+Cj2+CiCj=a. 2nCn+2b where a, b, n are even natural numbers and Cr represent the coefficient of xr in the expansion of (1+x)100 then which is/are CORRECT ?