Question

For r = 0, 1 ….., 10 if `A_(r ), B_(r )` and `C_(r )` denote respectively the coefficient of `X^(r )` in the expansions of `(1 + x)^(10), (1 + x)^(20)` and `(1 + x)^(30)`. Then `sum_(r=1)^(10) A_(r ) (B_(10) B_(r ) - C_(10) A_(r ))` is equal to
A. `B_(10) - C_(10)`
B. `A_(10) (B_(10)^(2) - C_(10)A_(10))`
C. 0
D. `C_(10) - B_(10)`

Answer 1

`A_(r)="Coefficient of "x^(r)" in "(1+x)^(10)=""^(10)C_(r)`
`B_(r)="Coefficient of "x^(r)" in "(1+x)^(20)=""^(20)C_(r)`
`C_(r)="Coefficient of "x^(r)" in"(1+x)^(30)=""^(30)C_(r)`
`:.underset(r=1)overset(10)sumA_(r)(B_(10)B_(r)-C_(10)A_(r))=underset(r=1)overset(10)sumA_(r)B_(10)B_(r)-underset(r=1)overset(10)sumA_(r)C_(10)A_(r)`
`=underset(r=1)overset(10)sum""^(10)C_(r)""^(20)C_(10)""^(20)C_(r)-underset(r=1)overset(10)sum""^(10)C_(r)""^(30)C_(10)""^(10)C_(r)`
`=underset(r=1)overset(10)sum""^(10)C_(10-r)""^(20)C_(10)""^(20)C_(r)-underset(r=1)overset(10)sum""^(10)C_(10-r)""^(30)C_(10)""^(10)C_(r)`
`=""^(20)C_(10)underset(r=1)overset(10)sum""^(10)C_(10-r).""^(20)C_(r)-""^(30)C_(10)underset(r=1)overset(10)sum""^(10)C_(10-r)""^(10)C_(r)`
`=""^(20)C_(10)(""^(30)C_(10)-1)-""^(30)C_(10)(""^(20)C_(10)-1)`
`=""^(30)C_(10)-""^(20)C_(10)=C_(10)-B_(10)`