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Recently I encountered a question but its answer as well as the way the author of the book has solved the question seemed wrong to me..

Find the sum of the coefficients of the expansion of

$$ (1+x-3x^2)^{2163} $$

I expanded the expression take it as $(1+(x-3x^2)$ and then I put the $(x-3x^2)$ as 1 thus leaving only the coefficients and hence putting the same values in the expression ( i.e. $ (x-3x^2)=1$) gave the answer as $2^{2163}$ but the answer according to the book is -1. The authors just put x=1 which seems absurd to me. Please help and tell me if am wrong.

Community
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Harsh Sharma
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    Hint: Imagine that the polynomial has already been multiplied out. How does plugging in $x=1$ add up the coefficients in that case? – Michael Burr May 26 '16 at 09:50

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Hint put $x=1$ as $1$ doesn't affect the binomial coefficients of an expansion.

Archis Welankar
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  • But if i expand binomially and them in the terms like (x-3x^2) switch x=1 that won't make the coefficients free from all the terms just like it does in (1+x)^n. Can you please elaborate. – Harsh Sharma May 26 '16 at 13:41
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    Go for multinomial theorem and see whether $x=1$ affects anything – Archis Welankar May 26 '16 at 14:57
  • Thanks for the help. Now i understood that if I switch x=1 in multinomial theorem, then what I get is simply the sum of coefficients. – Harsh Sharma May 28 '16 at 19:17