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Question: The coefficient of \(x^{5}\) in the expansion of \((1 + x^{2})^{5}(1 + x)^{4}\) is...

The coefficient of x5x^{5} in the expansion of (1+x2)5(1+x)4(1 + x^{2})^{5}(1 + x)^{4} is

A

30

B

60

C

40

D

None of these

Answer

60

Explanation

Solution

We have (1+x2)5(1+x)4(1 + x^{2})^{5}(1 + x)^{4} = (5C0+5C1x2+5C2x4+.....)(^{5} ⥂ C_{0} +^{5} ⥂ C_{1}x^{2} +^{5} ⥂ C_{2}x^{4} + .....)

(4C0+4C1x1+4C2x2+.......)(^{4} ⥂ C_{0} +^{4} ⥂ C_{1}x^{1} +^{4} ⥂ C_{2}x^{2} + .......)

So coefficient of x5x^{5} in [(1+x2)5(1+x)4]\lbrack(1 + x^{2})^{5}(1 + x)^{4}\rbrack

= 5C2.4C1+4C3.5C1=605C_{2}.^{4}C_{1} +^{4} ⥂ C_{3}.^{5}C_{1} = 60