Find the coefficient in the following expansion

Question

Find the coefficient of x⁹y⁶ in the expansion of (5x-2y)¹⁵

The coefficient of x⁹y⁶ is?

Answer 1

For an equation in the format (a+b)ⁿ we can find the (k+1)st term by the formula

 

[n!/[(k!)(n-k)!]]a^n-kb^k

 

In the equation given of (5x-2y)¹⁵ where we want the coefficient of the x⁹y⁶ term we have the following

 

n = 15, k = 6, a = 5x, b = -2y   so plugging into our factorial equation we get

 

[15!/((6!)(15-6)!)](5x)^15-6(-2y)⁶

 

[15!/(6!)(9!)](5x)⁹(-2y)⁶

 

5005(5)⁹x⁹(-2)⁶y⁶

 

5005(1953125)(64)x⁹y⁶

 

625,625,000,000x⁹y⁶

Answer 2

Check with Pascal's Triangle:        1    1

                                                1   2   1

                                              1  3    3   1

                                            1  4   6    4   1

 

 

Etc  until you get to the 15th row.  These are the coefficients to the (5x-2y)¹⁵ expansion.  Just the coefficients of the expansion.  You still have 5¹⁵ + 15(5¹⁴) + ...

 

I hope this helps.