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  • <img alt="\text { Find the coefficient of } x^6 \text { in }\left[(1+x)\left(1+x^2\right)^2\left(1+x^3\right)^3 \cdots\left(1+x^n\right)^n\right] \text {. }" src="https://entrancecorner.oncodecogs.

\text { Find the coefficient of } x^6 \text { in }\left[(1+x)\left(1+x^2\right)^2\left(1+x^3\right)^3 \cdots\left(1+x^n\right)^n\right] \text {. }

Option: 1

28


Option: 2

27


Option: 3

20


Option: 4

21


Answers (1)

best_answer

Taken mod x^7 ( meaning we drop anything with the power of x greater than 6), we have

\begin{gathered} (1+x)\left(1+x^2\right)^2\left(1+x^3\right)^3 \cdots \equiv(1+x)\left(1+2 x^2+x^4\right)\left(1+3 x^3+3 x^6\right)\left(1+4 x^4\right)\left(1+5 x^5\right)(1 \\ \left.+6 x^6\right) \\ \equiv\left(1+x+5 x^5+5 x^6\right)\left(1+2 x^2+5 x^4+8 x^6\right)\left(1+3 x^3+9 x^6\right) \end{gathered}

Where in second step we have paired (1+x) with 1+5x^5, \left(1+2 x^2+x^4\right) with \left(1+4 x^4\right), and \left(1+3 x^3+3 x^6\right) with \left(1+x^6\right). The coefficient of x^6 in the resulting expansion is 

5+8+9+1 \cdot 2 \cdot 3=28

\text { (where the } 1 \cdot 2 \cdot 3 \text { comes from the product } x \cdot 2 x^2 \cdot 3 x^3 \text { ). }

Posted by

Nehul

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