what-is-coefficient-of-t-3-in-the-expanssion-1-t-6-1-t-3-

Question Number 87074 by jagoll last updated on 02/Apr/20

what is coefficient of t^{3}

in the expanssion {\frac{1 - t^{6}}{1 - t}}^{3}

Commented by mr W last updated on 02/Apr/20

t^{2} : C_{2}^{4} = 6

t^{3} : C_{2}^{5} = 10

t^{4} : C_{2}^{6} = 15

t^{5} : C_{2}^{7} = 21

t^{6} : C_{2}^{8} - 3 = 25

Commented by mr W last updated on 02/Apr/20

t^{10} : C_{2}^{12} - 3 \times 15 = 21

Commented by jagoll last updated on 03/Apr/20

sir how get C_{2}^{5} ?

Commented by mr W last updated on 03/Apr/20

{(1 - t^{6})}^{3} h a s 1 o r t e r m s w i t h a t l e a s t t^{6}

s o w e o n l y n e e d t o s e e {(1 - t)}^{- 3} . t h e

c o e f . o f i t s t^{k} i s C_{3 - 1}^{k + 3 - 1} = C_{2}^{k + 2}, t h e r e f o r e

c o e f . o f t^{3} i s C_{2}^{3 + 2} = C_{2}^{5} .

Answered by malwaan last updated on 02/Apr/20

{\frac{1 - t^{6}}{1 - t}}^{3} =

{(1 + t + t^{2} + t^{3} + t^{4} + t^{5})}^{3} =

\Rightarrow (t^{3} + 3 t^{3} + 6 t^{3}) = 10 t^{3}

(t e r m s c o n t i n i n g t^{3} o n l y)

s o t h e c o e f f i c i e n t o f t^{3}

i s 10

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