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coefficient of x 2009 in the expansion of (1+x+x^2+x^3+x^4)^1001(1-x)^1002

coefficient of x 2009 in the expansion of (1+x+x^2+x^3+x^4)^1001(1-x)^1002

Grade:12

1 Answers

Aditya Gupta
2081 Points
4 years ago
the ques is not clear so kindly post a picture as well. however, i will try to answer the ques as is.
(1+x+x^2+x^3+x^4)^1001(1-x)^1002
=[(1+x+x^2+x^3+x^4)(1-x)]^1001(1-x)^1
= (1 – x^5)^1001*(1-x)
= (1 – x^5)^1001 – x(1 – x^5)^1001
in (1 – x^5)^1001, x^2009 cannot occur since 2009 is not divisible by 5.
in x(1 – x^5)^1001, to find coeff of x^2009, we need to find coeff of x^2008 in (1 – x^5)^1001, which will be zero since 2008 is not divisible by 5.
hence coeff is zero.
kindly approve :)

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