Find-coefficient-of-x-29-in-expansion-of-1-x-5-x-7-x-9-1000-

Question Number 161968 by naka3546 last updated on 24/Dec/21

F i n d c o e f f i c i e n t o f x^{29} i n e x p a n s i o n o f {(1 + x^{5} + x^{7} + x^{9})}^{1000} .

Answered by mr W last updated on 25/Dec/21

{(1 + x^{5} + x^{7} + x^{9})}^{1000}

= \sum_{a + b + c + d = 1000} (\begin{matrix} 1000 \\ a, b, c, d \end{matrix}) 1^{a} {(x^{5})}^{b} {(x^{7})}^{c} {(x^{9})}^{d}

= \sum_{a + b + c + d = 1000} (\begin{matrix} 1000 \\ a, b, c, d \end{matrix}) x^{5 b + 7 c + 9 d}

w i t h (\begin{matrix} 1000 \\ a, b, c, d \end{matrix}) = \frac{1000!}{a! b! c! d!}

f o r t e r m x^{29} : 5 b + 7 c + 9 d = 29

d = 0, c = 2, b = 3, a = 995

d = 1, c = 0, b = 4, a = 995

c o e f . o f x^{29} i s t h e r e f o r e

(\begin{matrix} 1000 \\ 0, 2, 3, 995 \end{matrix}) + (\begin{matrix} 1000 \\ 1, 0, 4, 995 \end{matrix})

= \frac{1000!}{995!} (\frac{1}{0! 2! 3!} + \frac{1}{1! 0! 4!})

= \frac{1000!}{995!} \times (\frac{1}{12} + \frac{1}{24})

= \frac{1000!}{8 \times 995!}

= \frac{1}{8} \times 1000 \times 999 \times 998 \times 997 \times 996

= 123 754 368 753 000

Commented by naka3546 last updated on 25/Dec/21

T h a n k y o u, s i r .

Commented by Tawa11 last updated on 25/Dec/21

Great sir

Commented by Jamshidbek last updated on 25/Dec/21

Which is this theorem ?

Commented by mr W last updated on 25/Dec/21

b i n o m i a l t h e o r e m

Commented by mr W last updated on 25/Dec/21

Commented by Jamshidbek last updated on 25/Dec/21

thank you