Question

$\sum_{\substack{i,j=0 \\\ t \neq j}}^n$ $^nC_i\ ^nC_j$
is equal to

• A. $2^{2n \text\\_2n}C_n$
• B. $2^{2n-1\\_2n-1}C_{n-1}$
• C. $2^{2n-\frac{1}{2}}\ ^{2n}C_n$
• D. $2^{n-1}+2^{2n-1}C_n$

Answer 1

Answer: (A)

$2^{2n \text\\_2n}C_n$

Answer 2

The correct answer is (A) : $2^{2n \text\\_2n}C_n$
$\sum_{\substack{i,j=0 \\\ t \neq j}}^n$ $^nC_i\ ^nC_j$
$= ∑^{n}_{i,j = 0}$ $^nC_i\ ^nC_j - ∑^{n}_{i=j}\ ^nC_i\ ^nC_j$
$= ∑^{n}_{j=0}\ ^nC_i ∑^{n}_{j =0}\ ^nC_j - ∑^{n}_{ i =0}\ ^nC_i\ Ci$
$= 2^n.2^n-\ ^{2n}C_n$
$= 2^{2n\\_2n}C_n$