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Question

Find the coefficient of:
(i) x in the expansion of (12x3+3x5)(1+1x)8

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Solution

Coeff of x in (12x3+3x5)(1+1x)8

= coeff of x in (1+1x)8+ coeff of x in (2x3)(1+1x)8+ coeff of x in 3x5(1+1x)8

= coeff of x in (1+1x)82 coeff of x2 in (1+1x)8+3 coeff of x4 in (1+1x)8

rth term in (1+1x)8

$t_r = {^8C}_{r-1} ( \frac{1}{x})^{r-1} = {^8C_{r-1} x ^{1-r} $

For 1r=1,r=0, coeff of x=0

For 1r=2,r=3, coeff of x=8C2

For 1r=4,r=5, coeff of x=8C4

Therefore coeff of x=028C2+3×8C4
=154


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