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({:(30),(0):})({:(30),(10):})-({:(30),(1...

`({:(30),(0):})({:(30),(10):})-({:(30),(1):})({:(30),(11):}) + ... + ({:(30),(20):})({:(30),(30):}) ` is equal to

A

`.^(30)C_(11)`

B

`.^(60)C_(10)`

C

`.^(30)C_(10)`

D

`.^(65)C_(55)`

Text Solution

Verified by Experts

`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)`
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