If a and b are coefficients of $x^{n}$ in the expansion of $(1 + x)^{2 n}$ and $(1 + x)^{2 n - 1}$ respectively, then write the relation between a and b.

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Solution

Coefficient of $x^{n}$ in the expansion $(1 + x) 2 n = 2 n C_{n} = a$

Coefficient of x^n in the expansion $(1 + x) 2 n - 1 = {2 n - 1}_{C_{n}} = b$

Now, we have :

$2 n_{C_{n}} = \frac{2 n!}{n! n!} = \frac{2 n (2 n - 1)!}{n (n - 1)! n!} . . . (i)$

and $2 n_{1} C_{n} = \frac{(2 n - 1)!}{n! (n - 1)!} . . . (i i)$

Dividing equation (1) by (2), we get

$\Rightarrow \frac{2 n_{C_{n}}}{2 n - 1_{C_{n}}} = \frac{2 n (2 n - 1)! n! (n - 1)!}{n (n - 1)! n! (2 n - 1)!}$

$\Rightarrow \frac{a}{b} = 2$

$\Rightarrow a = 2 b$

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