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Question
The coefficient of xn−1 in the polynomial (x+2n+1C0)(x+2n+1C1)(x+2n+1C2)⋯(x+2n+1Cn) is
The coefficient of xn−1 in the polynomial (x+2n+1C0)(x+2n+1C1)(x+2n+1C2)⋯(x+2n+1Cn) is
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Solution
The correct option is C 22n
(x+2n+1C0)(x+2n+1C1)(x+2n+1C2)+...(x+2n+1Cn)
=xn+[2n+1C0+2n+1C1+...2n+1Cn]xn−1+...
Hence
Coefficient of xn−1
=2n+1C0+2n+1C1+...2n+1Cn
=12[2(2n+1C0+2n+1C1+...2n+1Cn)]
=12[2n+1C0+2n+1C1+...2n+1Cn+2n+1Cn+1+2n+1Cn+2+...2n+1C2n+1]
Since nCr=nCn−r
Now
12[2n+1C0+2n+1C1+...2n+1Cn+2n+1Cn+1+2n+1Cn+2+...2n+1C2n+1]
=12[(1+x)2n+1]x=1
=12[22n+1]
=22n.
(x+2n+1C0)(x+2n+1C1)(x+2n+1C2)+...(x+2n+1Cn)
=xn+[2n+1C0+2n+1C1+...2n+1Cn]xn−1+...
Hence
Coefficient of xn−1
=2n+1C0+2n+1C1+...2n+1Cn
=12[2(2n+1C0+2n+1C1+...2n+1Cn)]
=12[2n+1C0+2n+1C1+...2n+1Cn+2n+1Cn+1+2n+1Cn+2+...2n+1C2n+1]
Since nCr=nCn−r
Now
12[2n+1C0+2n+1C1+...2n+1Cn+2n+1Cn+1+2n+1Cn+2+...2n+1C2n+1]
=12[(1+x)2n+1]x=1
=12[22n+1]
=22n.
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