Find the sum of the products, two at a time, of the coefficients in the expansion of $(1 + x)^{n}$ when $n$ is a positive integer.

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Solution

$(1 + x)^{n} = C_{0} + C_{1} x + C_{2} x^{2} . . . . . C_{n} x^{n}$ .............(1)

squaring both sides and put x = 1

$(2)^{2 n} = [C_{0} + C_{1} + C_{2} . . . . . C_{n}]^{2}$

$(2)^{2 n} = [C_{0}^{2} + C_{1}^{2} + C_{3}^{2} + . . . . . . . C_{n}^{2}] + 2. [C_{1} C_{2} + C_{2} C_{3} . . . . .]$

let us assume sum taken two at a time be S

$S = \frac{(2)^{2 n} - [C_{0}^{2} + C_{1}^{2} + C_{3}^{2} + . . . . . . . C_{n}^{2}]}{2}$

replace $x$ by $\frac{x}{2}$ from eqn (1)

and multiply with eqn (1)

$(1 + x)^{n} . (1 + \frac{1}{x})^{n} = (1 + x)^{2 n} . \frac{1}{x^{n}} = [C_{0} + C_{1} x + C_{2} x^{2} . . . . . C_{n} x^{n}] . [C_{0} + C_{1} \frac{1}{x} + C_{2} \frac{1}{x^{2}} + . . . . . C_{n} \frac{n}{x^{n}}]$

contant term at $R H S = C_{0}^{2} + C_{1}^{2} + C_{3}^{2} + . . . . . . . C_{n}^{2}$

contant term at $L H S$ = coefficient of $x^{n}$ = constant term at RHS

$= C_{n}^{2 n} = C_{0}^{2} + C_{1}^{2} + C_{3}^{2} + . . . . . . . C_{n}^{2}$

using above value

$S = \frac{(2)^{2 n} - C_{n}^{2 n}}{2}$

$= 2^{2 n - 1} - \frac{(2 n)!}{2. n! . n!}$

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