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Question

The coefficient of the term independent of x in the expansion of (1+x+2x3)(32x213x)9

A
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B
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C
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D
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Solution

The correct option is C
The general term in the expansion of (32x213x)9 is
T(r+1)=9Cr(32x2)9r(13x)r=9Cr(32)9r(13)x183r
Now, the coefficient of the term independent of x in the expansion of (1+x+2x3)(32x213x)9 = Sum of the coefficient of the terms x0,x1 and x3 in (32x213x)9 .....(ii)
For x0 in (i) above, 183r=0r=6.x1 in (i) above, there exists no value of r and hence no such term exits. For x3 in (i), 183r=3r=7
For term independent of x, in (ii) the coefficient
=1×9C6(1)6(32)96(13)6+2×9C7(1)7(32)97(13)7
=9.8.71.2.3.3323.136+29.81.2(1)3222.137=718227=1754

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