Binomial Theorem

Let , , and be the coefficients of x7,x5,x3 and x respectively in the expansion of (x+ x3-1 )5+ (x- x3-1 )5, x>1. If u and v satisfy the equations u+ v=18, u+ v=20, then u+v equals :

1. 4
2. 3
3. 5
4. 8

Answer

(3) 5

Solution

(a+b)5+(a-b)5=2 [a5+10a3b2+5ab4 ] (odd powers of b cancel), a=x, b2=x3-1

=10x7+20x6+2x5-20x4-20x3+10x =10, =2, =-20, =10

10u+2v=18, -20u+10v=20 u=1, v=4

u+v=5

Answer: (3)