SOLUTION: I need help on, find the coefficient of x^15 in the expansion of (1-x^2+x^3)^20

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Question 1146065: I need help on, find the coefficient of x^15 in the expansion of
(1-x^2+x^3)^20
Answer by greenestamps(13198) About Me  (Show Source):
You can put this solution on YOUR website!


You will get x^15 terms with the following combinations of terms:

(1) %28x%5E3%29%5E5%2A%281%29%5E15
(2) %28x%5E3%29%5E3%2A%28-x%5E2%29%5E3%2A%281%29%5E14
(3) %28x%5E3%29%5E1%2A%28-x%5E2%29%5E6%2A%281%29%5E13

The numbers of ways to get each of these combinations are
(1) 20%21%2F%28%285%21%29%2815%21%29%29+=+15504
(2) 20%21%2F%28%283%21%29%283%21%29%2814%21%29%29+=+775200
(3) 20%21%2F%28%281%21%29%286%21%29%2813%21%29%29+=+542640

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

15504-775200%2B542640+=+-217056

You can confirm this answer by entering your expression in the box at wolframalpha.com.