SOLUTION: Find the coefficient of x^27 in the expansion of (x^2+3x)^15

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Question 1185340: Find the coefficient of x^27 in the expansion of (x^2+3x)^15
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
Every term of %28a%2Bb%29%5En is of the form %28matrix%282%2C1%2Cn%2Cr%29%29a%5Er%2Ab%5E%28n-r%29

Therefore, every term of %28x%5E2%2B3x%29%5E15 is of the form

%28matrix%282%2C1%2C15%2Cr%29%29%28x%5E2%29%5Er%2A%283x%29%5E%2815-r%29

%28matrix%282%2C1%2C15%2Cr%29%29x%5E%282r%29%2A3%5E%2815-r%29%2Ax%5E%2815-r%29

%28matrix%282%2C1%2C15%2Cr%29%29%2A3%5E%2815-r%29x%5E%2815%2Br%29

We want the exponent of x to be 27, so we set

15+r = 27
   r = 12

%28matrix%282%2C1%2C15%2C12%29%29%2A3%5E%2815-12%29x%5E%2815%2B12%29

455%2A3%5E%283%29x%5E%2827%29

455%2A27x%5E%2827%29

12285x%5E%2827%29

Its coefficient is 12285

Edwin