In the expansion $(1 + x)^{5} = 1 + 5 x + a x^{2} . . . . . . x^{5}$ , find the value of a.

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Solution

We can solve this if we know the binomial expansion of $(1 + x)^{5}$
.

Binomial expansion of $(a + b)^{n} i s (a + b)^{n} = a^{n} + n c_{1} a^{n - 1} b + n c_{2} a^{n - 2} b^{2} + . . . . b^{n}$

$\Rightarrow (1 + x)^{5} = 1 + 5 c_{1} x + 5 c_{2} x^{2} . . . . . . .5 c_{5} x^{5}$

$\Rightarrow a = 5 c_{2} = \frac{5!}{3! \times 2!} = 10$

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