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The coefficient of t^4 in ((1-t^6)/(1-t)...

The coefficient of `t^4` in `((1-t^6)/(1-t))^3`

A

12

B

10

C

15

D

14

Text Solution

Verified by Experts

The correct Answer is:
C
Clearly, `((1 - t^(6))/( 1 - t))^(2) = (1 - t^(6))^(3) (1 - t)^(-3)`
`:.` Coefficient of `t^(4)` in `(1 - t^(6))^(3) (1 - t)^(-3)`
= Coefficient of `t^(4)` in `(1 - t^(18)) - 3t^(6) + 3 t^(12)) (1 - t)^(-3)`
= Coefficient of `t^(4)` in `(1 - t)^(-3)`
`= .^(3 + 4 - 1)C_(4) = .^(6)C_(4) = 15`
(`:'` coefficient of `x^(r)` in `(1 - x)^(-n) = .^(n + r + 1)C_(r )`)
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