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  4. The coefficient of xn in the expansion of (1 + x)2n and (1 + x)2n-1 are in the ratio

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Q. The coefficient of $x^n$ in the expansion of $(1 + x)^{2n}$ and $(1 + x)^{2n-1}$ are in the ratio

Binomial Theorem

Solution:

Coefficient of $x^n$ in $\left(1+x\right)^{2n} = \,{}^{2n}C_{n}$
and coefficient of $x^{n}$ in $\left(1 + x\right)^{2n-1}= \,{}^{2n-1}C_{n}$
$\therefore $ Required ratio $= \frac{^{2n}C_{n}}{^{2n-1}C_{n}}$
$= \frac{\frac{\left(2n\right)!}{n!n!}}{\frac{\left(2n-1\right)!}{n!\left(n-1\right)!}} = 2 : 1$