If A -2 n C 0-2 n C 1-2 n C 12 n -1 C 1-2 n C 22 n -2 C 1-2 n C 2 n -1-1 C 1then A isA- n - 22 nB- 0C- 1D- 2 n

The correct option is A n ·2^2 nA= Coefficient of x in [^2 n C_0(1+x)^2 n. .+^2 n C_1(1+x)^2 n 1+… . .^2 n C_2n 1(1+x)^1] = Coefficient of x in [(1+(1+x)]^2 n ...

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Standard XII

Mathematics

Binomial Expression

If A =2 n C 0...

Question

If $A =^{2 n} C_{0} \cdot^{2 n} C_{1} +^{2 n} C_{1}^{2 n - 1} C_{1} +^{2 n} C_{2}^{2 n - 2} C_{1} + \dots +^{2 n} C_{2 n - 1} \cdot^{1} C_{1}$
then $A$ is

0

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2^{n}

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n \cdot 2^{2 n}

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Solution

The correct option is C $n \cdot 2^{2 n}$
$A =$ Coefficient of $x$ in $[^{2 n} C_{0} (1 + x)^{2 n}$
$+^{2 n} C_{1} (1 + x)^{2 n - 1} + \dots . .^{2 n} C_{2 n - 1} (1 + x)^{1}]$
$=$ Coefficient of $x$ in $[(1 + (1 + x)]^{2 n} -^{2 n} C_{2 n}$
$=$ Coefficient of $x$ in $(2 + x)^{2 n}$
$=$ Coefficient of $x$ in $2^{2 n} {(1 + \frac{x}{2})}^{2 n} = n \cdot 2^{2 n}$

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Similar questions

How many of the following statements are true?
(1) $r^{n} C_{r} = n \times^{n - 1} C_{r - 1}$

(2) Each term of the sum
$(^{2 n} C_{1})^{2} + 2 \times (^{2 n} C_{2})^{2} + 3. (^{2 n} C_{3})^{2} . . . . . + 2 n . (^{2 n} C_{2 n})^{2}$ can be written as $r (^{2 n} C_{r})^{2}$
(3) coefficient of $x^{2 n - 1}$ in
$(^{2 n - 1} C_{0} +^{2 n - 1} C_{1} x +^{2 n - 1} C_{2} x^{2} . . . . .^{2 n - 1} C^{2 n - 1} x^{2 n - 1}) \times (^{2 n} C_{0} +^{2 n} C_{1} x +^{2 n} C_{2} x^{2} . . . . .^{2 n} C_{2 n} x^{2 n})$
= $^{2 n - 1} C_{0} \times^{2 n} C_{2 n - 1} +^{2 n - 1} C_{1} \times^{2 n} C_{2 n - 2} . . . . .^{2 n - 1} C_{2 n - 1} .^{2 n} C_{0}$

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The value of $2 (^{n} C_{0}) + \frac{3}{2} (^{n} C_{1}) + \frac{4}{3} (^{n} C_{2}) + \frac{5}{4} (^{n} C_{3}) + \dots$ is

Q. The value of

\frac{1}{2}^{n} C_{1} .^{n - 1} C_{1} + \frac{2}{3}^{n} C_{2} + \frac{3}{4}^{n} C_{c} .^{n - 1} C_{3} + \dots \dots

Find $^{n} C_{0}$ - 2 $^{n} C_{1}$ + 3 $^{n} C_{2}$ - 4 $^{n} C_{3}$ .....................+ $(- 1)^{n}$ (n+1) $^{n} C_{n}$

Find the value of $^{n} C_{1} + 2^{n} C_{2} + 3^{n} C_{3} . . . . . . . . . n^{n} C_{n}$

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Binomial Expression

Standard XII Mathematics

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If A=2nC0⋅ 2nC1+ 2nC1 2n−1C1+ 2nC2 2n−2C1+…+2nC2n−1⋅ 1C1 then A is

The correct option is C n⋅22nA= Coefficient of x in [2nC0(1+x)2n +2nC1(1+x)2n−1+…..2nC2n−1(1+x)1] = Coefficient of x in [(1+(1+x)]2n−2nC2n = Coefficient of x in (2+x)2n = Coefficient of x in 22n(1+x2)2n=n⋅22n

If $A =^{2 n} C_{0} \cdot^{2 n} C_{1} +^{2 n} C_{1}^{2 n - 1} C_{1} +^{2 n} C_{2}^{2 n - 2} C_{1} + \dots +^{2 n} C_{2 n - 1} \cdot^{1} C_{1}$
then $A$ is