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 `underset(r=1)overset(10)sum A_(r)(B_(10)B_(r ) - C_(10)A_(r ))` is equal to

Options

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

Answer

d

Explanation

We have,
`A_(r) = ""^(10)C_(r), B_(r) = ""^(20)C_(r) and C_(r) = ""^(30)C_(r)`
`because sum _(r=1)^(10) A_(r) (B_(10)B_(r)- C_(10)A_(r))`
`= B_(10) sum_(r=1)^(10) A_(r) B_(r) - C_(10) sum_(r=1)^(10) (A_(r))^(2)`
`= B_(10)(sum_(r=1)^(10) ""^(10)C_(r) ""^(20)C_(r)) - C_(10) sum_(r=1)^(10) (""^(10)C_(r))^(2)`
`= B_(10)(sum_(r=1)^(10) ""^(10)C_(r) ""^(20)C_(20-r)) - C_(10) {sum_(r=1)^(10) (""^(10)C_(r))^(2) -1}`
`= B_(10){sum_(r=1)^(10) ""^(10)C_(r) ""^(20)C_(20-r)-1} - C_(10) {sum_(r=1)^(10) (""^(10)C_(r))^(2) -1}`
` b_(10)XX` {Coefficient of `x^(20)` in `(1 + x )^(10) (x + 1) ^(20) -1} - C_(10) {""^(20)C_(10 -1)}`
`B_(10)xx {""^(30) C_(20) -1} - C_(10) {""^(20)C_(10) -1}`
`B_(10){ ""^(30)C_(10) -1} - C_(10) {""^(20)C_(10) -1} `
`B_(10) (C_(10) -1} - C_(10) { ""^(20) C_(10) -1}`.