Find the 7th term of the expansion (x+y)^21

Question

step by step solution please :)

Answer 1

Binomial Theorem says.....

1st term is (21 choose 0) x^21 y^0

2nd term is (21 choose 1) x^20 y

3rd term is (21 choose 2) x^19 y^2

etc etc etc.... <--- what is the pattern???

The pattern is, IN GENERAL, for the Kth term...

(21 choose (k-1)) x^(21-k+1) y^(k-1)

7th term is (21 choose 6) x^15 y^7

(21 choose 6) = (21*20*19*18*17*16*15! / (6! 15!))

= (21*20*19*18*17 *16)/(6*5*4*3*2) <-- 15! cancels

= (21 *2*19*17*4) <--- 18 cancels 6*3; 20 cancels 5*2; 16 cancels 4

= 54264

so the 7th term is 54264 x^15 y^6