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((30),(0)) ((30),(10))- ((30),(1)) ((30)...

`((30),(0)) ((30),(10))- ((30),(1)) ((30),(11))+ ((30),(2))((30),(12))+......((30),(20))((30),(30))` is equal

A

`((30),(10))`

B

`((30),(15))`

C

`((60),(30))`

D

`((31),(10))`

Text Solution

Verified by Experts

The correct Answer is:
A
`(1+x)^30 =.^30C_0 + .^30C_1x + .^30C_2x^2`+….. + `.^30C_20x^20`+…..+ `.^30C_30x^30`…(i)
`(x-1)^30=.^30C_0 x^30-.^30C_1x^29`+….+`.^30C_10x^20 - .^30C_11x^19+ .^30C_12x^18`+…..+ `.^30C_30`…(ii)
Multiplying equation (i) and (ii) and comparing coefficient of `x^20` , then we get :
`(.^30C_0)(.^30C_10)-(.^30C_1)(.^30C_11)+(.^30C_2)(.^30C_12)` +....+ `(.^30C_20)(.^30C_30)=.^30C_10`
[`because (1+x)^30 (x-1)^30=(x^2-1)^30=(1-x^2)^30` and coefficient of `x^20=.^30C_10`]
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