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Ratio and Proportion

If the coeffi...

Question

If the coefficient of $x^{n}$ in $(1 + x)^{2 n}$ is ' $a$ ' and the coefficient of $x^{n}$ in $(1 + x)^{2 n - 1}$ is $b$ , then $\frac{a}{b} =$

2

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4

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2 n

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n

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Solution

The correct option is A $2$
The coefficient of $x^{n}$ in $(1 + x)^{2 n}$
$=^{2 n} C_{n} = a$ ...(i)
The coefficient of $x^{n}$ in $(1 + x)^{2 n - 1}$
$=^{2 n - 1} C_{n} = b$ ...(ii)

Hence, $\frac{a}{b}$ $= \frac{^{2 n} C_{n}}{^{2 n - 1} C_{n}}$

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

$= \frac{2 n}{n}$

$= 2$

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