The value of `((30), (0))((30), (10))-((30), (1))((30),( 11)) +((30), (2))((30), (12))+.....+((30), (20))((30), (30))=` `^60 C 20` b. `^30 C 10` c. `^60 C 30` d. `^40 C 30`

Options

A. `.^(30)C_(11)`
B. `.^(60)C_(10)`
C. `.^(30)C_(10)`
D. `.^(65)C_(55)`

Explanation

`A={:((30),(0)):}{:((30),(10)):}-{:((30),(1)):}{:((30),(11)):}+{:((30),(2)):}{:((30),(12)):}-...+{:((30),(20)):}{:((30),(30)):}`
`:.A=""^(30)C_(0).""^(30)C_(10)-""^(30)C_(1).""^(30)C_(11)+""^(30)C_(2).""^(30)C_(12)-...+""^(30)C_(20).""^(30)C_(30)`
`="Coefficient of "x^(20)" in "(1+x)^(30)(1-x)^(30)`
`="Coefficient of "x^(20)" in "(1-x^(2))^(30)`
`="Coefficient of "x_(20)" in "underset(r=0)overset(30)sum(-1)^(r^(30))C_(r)(x^(2))^(r)`
`=(-1)^(10)""^(30)C_(10)" "["for coefficient of "x^(20)," put "r=10]`
`=""^(30)C_(10)`