For `r = 0, 1,"…..",10`, let `A_(r),B_(r)`, and `C_(r)` denote, respectively, the coefficient of `x^(r )` in the expansion 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

Options

A. `B_(10) - C_(10)`
B. `A_(1)(B_(10)^(2) - C_(10)A_(10))`
C. 0
D. `C_(10) - B_(10)`

Answer

D

Explanation

`A_(r), B_(r)`, and `C_(r)` denotes, respectively, the coefficient of `x^(r)` in the expansion of `(1+x)^(10), (1+x)^(20)` and `(1+x)^(30)`.
`:. A_(r) = .^(10)C_(r),B_(r)=.^(20)C_(r),C_(r)=.^(30)C_(r)`
`:. underset(r=1)overset(10)sumA_(r)(B_(10)B_(r)-C_(10)A_(r))`
`= B_(10)underset(r=1)overset(10)sumA_(r)B_(r)-C_(10)underset(r=1)overset(10)sum(A_(r))^(2)`
`= B_(10)underset(r=1)overset(10)sum.^(10)C_(r).^(20)C_(r)-C_(10)underset(r=1)overset(10)sum(.^(10)C_(r))^(2)`
`=B_(10)underset(r=1)overset(10)sum.^(10)C_(r).^(20)C_(20-r)-C_(10)underset(r=1)overset(10)sum(.^(10)C_(r))^(2)`
`= B_(10)[(underset(r=0)overset(10)sum.^(10)C_(r).^(20)C_(20-r))-1]-C_(10)[(underset(r=0)overset(10)sum(.^(30)C_(r))^(2))-1]`
`=B_(10)[.^(30)C_(2)-1]-C_(10)[.^(20)C_(10) - 1]`

`( :'.^(n)C_(0)^(2)+.^(n)C_(1)^(2)+.^(n)C_(2)^(2)+"...."+.^(n)C_(n)^(2)=.^(2n)C_(n))`
`=[B_(10).^(30)C_(20)-C_(10).^(20)C_(10)]+[C_(10)-B_(10)]`
`=[.^(20)C_(10).^(30)C_(20)-.^(30)C_(10).^(20)C_(10)] + [C_(10) - B_(10)]`
`= C_(10) - B_(10)`