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Question

The term independent of x expansion of (x+1x23x13+1x1xx12)10 is

A
4
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B
120
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C
210
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D
310
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Solution

The correct option is C 210
[x+1x23x13+1x1xx12]10
=⎢ ⎢ ⎢(x13)3+13x23x13+1{(x)21}x(x1)⎥ ⎥ ⎥10
=⎢ ⎢ ⎢(x13+1)(x23+1x13)x23x13+1{(x)21}x(x1)⎥ ⎥ ⎥10
[(x13+1)(x+1)x]10=(x13x12)10
The general term is
Tr+1=10Cr(x13)10r(x12)r=10Cr(1)rx10r3r2
For independent of x, put
10r3r2=0
20 - 2r - 3r = 0
20 = 5r r = 4
T5=10C4=10×9×8×74×3×2×1=210

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