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

1 : 2

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1 : 3

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3 : 1

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2 : 1

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Solution

The correct option is D 2 : 1
$C o e f f i c i e n t s o f x^{n} i n (1 + x)^{2 n} a n d (1 + x)^{2 n - 1} a r e^{2 n} C_{n} a n d^{2 n - 1} C_{n} r e s p e c t i v e l y . ∴ N o w, \frac{2 n_{c_{n}}}{2 n - 1_{c_{n}}} = \frac{(2 n)!}{n! n!} \frac{n! (n - 1)!}{(2 n - 1)!} = \frac{2 n}{n} = 2 ∴^{2 n} C_{n} :^{2 n - 1} C_{n} = 2 : 1$

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Similar questions

Q. Prove that the coefficient of

x^{n}

in the expansion of

(1 + x)^{2 n}

is twice the coefficient of

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in the expansion of

(1 + x)^{2 n - 1}

Q. If A and B are the coefficient of xⁿ in the expansion of (1 + x)²ⁿ and (1 + x)^{2n – 1} respectively, then

\frac{A}{B} =

___________.

Prove that the coefficient of $x^{n}$ in the binomial expansion of $(1 + x)^{2 n}$ is twice the coefficient of $x^{n}$ in the binomial expansion of $(1 + x)^{2 n - 1}$ .

Q. If a and b are the coefficients of xⁿ in the expansion of

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Q. If A and B are the coefficients of

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and

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respectively, then

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

The correct option is D 2 : 1Coefficients of xn in (1+x)2n and(1+x)2n−1 are 2nCn and 2n−1Cn respectively.∴ Now,2ncn2n−1cn=(2n)!n!n!n!(n−1)!(2n−1)!=2nn=2∴ 2nCn:2n−1Cn=2:1

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