The value of`sum_(r=0)^(10) (-1)^(r).4^(10-r)""^(30)C_(r)""^(30-r)C_(10-r)` is equal to

Options

A. `.^(30)C_(10) xx 2^(10)`
B. `.^(30)C_(9) xx 4^(10)`
C. `.^(30)C_(10) xx 3^(10)`
D. `.^(30)C_(9) xx 4^(10)`

Answer

C

Explanation

`underset(r=0)overset(10)sum(-1)^(r).4^(10-r)..^(30)C_(r)..^(30-r)C_(10-r)`
` = underset(r=0)overset(10)sum(-1)^(r)..4^(10-r).(30!)/((30-r)!r!) .((30-r)!)/(20!(10-r)!)`
`= (30!)/(20!)underset(r=0)overset(10)sum(-1)^(r).4^(10-r).(1)/(r!(10-r)!)`
`= (30!)/(10! xx 20!)underset(r=0)overset(10)sum(-1)^(r).4^(10-r).(10!)/(r!(10-r)!)`
`= .^(30)C_(10)underset(r=0)overset(10)sum(-1)^(r).4^(10-r)..^(10)C_(r)`
`= .^(30)C_(10)(4-1)^(10)`
`= .^(30)C_(10).3^(10)`