If `(1+x)^n=C_(0)C_1c+C_(2)x^2+…..+C_(n)x^n` then show that the sum of the products of the `C_(i)` taken two at a time represented by :`Sigma_(0 le I lt) Sigma_( j le n) C_(i)C_(j)` "is equal to " 2^(2n-1)-(2n!)/(2.n! n !)`

Explanation

To solve the problem, we need to show that the sum of the products of the coefficients \( C_i \) taken two at a time is equal to \( 2^{2n-1} - \frac{(2n)!}{(2.n)! n!} \).

### Step 1: Understand the coefficients \( C_i \)

The coefficients \( C_i \) in the expansion of \( (1+x)^n \) are given by the binomial theorem:
\[ C_i = \binom{n}{i} \]
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