The coefficient of t4 in the expansion 1−t61−t3, is

1−t61−t3=1−t63(1−t)−3=1−3t6+3t12−t18(1−t)−3=(1−t)−3−3t6(1−t)−3+3t12(1−t)−3−t18(1−t)−3Coefficient of t4 in 1−t61−t3= Coefficient of t4 in (1−t)−3=4+3−1C3−1 [ ∵ Coeff. of xn in (1−x)−r=n+r−1Cr−1 ] =6C2=15

The coefficient of t4 in the expansion 1−t61−t3, is

1−t61−t3=1−t63(1−t)−3=1−3t6+3t12−t18(1−t)−3=(1−t)−3−3t6(1−t)−3+3t12(1−t)−3−t18(1−t)−3Coefficient of t4 in 1−t61−t3= Coefficient of t4 in (1−t)−3=4+3−1C3−1 [ ∵ Coeff. of xn in (1−x)−r=n+r−1Cr−1 ] =6C2=15

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The coefficient of $t^{4}$ in the expansion ${(\frac{1 - t^{6}}{1 - t})}^{3},$ is

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Detailed Solution

${(\frac{1 - t^{6}}{1 - t})}^{3}$

$\begin{aligned} = {(1 - t^{6})}^{3} (1 - t)^{- 3} = (1 - 3 t^{6} + 3 t^{12} - t^{18}) (1 - t)^{- 3} \\ = (1 - t)^{- 3} - 3 t^{6} (1 - t)^{- 3} + 3 t^{12} (1 - t)^{- 3} - t^{18} (1 - t)^{- 3} \end{aligned}$