Question 1146065
<br>
You will get x^15 terms with the following combinations of terms:<br>
(1) {{{(x^3)^5*(1)^15}}}
(2) {{{(x^3)^3*(-x^2)^3*(1)^14}}}
(3) {{{(x^3)^1*(-x^2)^6*(1)^13}}}<br>
The numbers of ways to get each of these combinations are
(1) {{{20!/((5!)(15!)) = 15504}}}
(2) {{{20!/((3!)(3!)(14!)) = 775200}}}
(3) {{{20!/((1!)(6!)(13!)) = 542640}}}<br>
The signs of (1) and (3) are positive; the sign of (2) is negative.  So the coefficient of x^15 in (1-x^2+x^3)^20 is<br>
{{{15504-775200+542640 = -217056}}}<br>
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