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Question

Mathematics Question on binomial expansion formula

The co-efficient of x3x^3 in the expansion of (1−x+x2)5(1-x+x^2)^5 is

A

10

B

-20

C

-50

D

-30

Answer

-30

Explanation

Solution

No. of terms in (1−x+x2)5=1+x(x−1)5\left(1 - x+ x^{2}\right)^{5} = 1 + x\left(x- 1\right)^{5} = 5C0+ 5C1x(x−1)+ 5C2x2(x−1)2= \,^{5}C_{0} + \,^{5}C_{1}x\left(x-1\right)+ \,^{5}C_{2}x^{2}\left(x-1\right)^{2} + 5C3x3(x−1)3+ 5C4x4(x−1)4+ 5C5x5(x−1)5+ \,^{5}C_{3}x^{3} \left(x-1\right)^{3}+ \,^{5}C_{4}x^{4}\left(x-1\right)^{4}+ \,^{5}C_{5}x^{5}\left(x-1\right)^{5} ∴\therefore co - eff. x3=−2⋅5C2− 5C3=−30x^{3} = - 2\cdot^{5}C_{2}- \,^{5}C_{3} = -30