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Question

The expression C0n+2C1n+3C2n+n+1Cnn is equal to


A

n+12n

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B

2nn+2

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C

n+22n-1

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D

n+22n+1

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Solution

The correct option is C

n+22n-1


Simplify the given expression:

The given equation is C0n+2C1n+3C2n+n+1Cnn.

As we know, 1+xn=C0n+x.C1n+x2.C2n+....xn.Cnn.

Multiply x on both sides of the equation,

x1+xn=x.C0n+x2.C1n+x3.C2n+....xn+1.Cnn

Differentiate the equation w.r.t. to x,

1+xn+nx1+xn-1=C0+2x.C1n+3x2.C2n+....n+1.Cnn

Put x=1in the above expression,

⇒2n+n.2n-1=C0n+2C1n+3C2n+....n+1.Cnn=2n-1n+2

Hence, the given expression can be simplified to n+22n-1 and therefore option C is correct.


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