For x 2y10 what is the coefficient of x3y7 A 120 B 120 C 96

For (x -2y)¹⁰ , what is the coefficient of x³y⁷?

A. +120

B. -120

C. -960

D. +960

E. none of the above

Solution

The triangle gives the coefficients when a binomial is raised to power n. The numbers for power n are obtained by adding numbers in power n-1.

n=0: 1
n=1: 1 1
n=2: 1 2 1
n=3: 1 3 3 1
n=4: 1 4 6 4 1
n=5: 1 5 10 10 5 1
n=6: 1 6 15 20 15 6 1
n=7: 1 7 21 35 35 21 7 1
n=8: 1 8 28 56 70 56 28 8 1
n=9: 1 9 36 84 126 126 84 36 9 1
n=10: 1 10 45 120 210 256 210 120 45 10 1

Now, (x+y)10 = _x10 + _x9y + _x8y2 + ... + _x3y7 + _x2y8 + _xy9 + _y10
where _ denotes the coefficients.

Looking at the coefficients for n=10, we see that the answer is 120.

There is another way to do this. By binomial theorem we know that
(x+y)^n = (n,0)(x^n)(y^0) + (n,1)(x^n-1)(y^1) + ...

Since n=10, the coefficient for x3y7 will be (10,7) = 10! 7! / (10-7)! = 10*9*8/3*2 = 720/6 = 120