what-is-coefficient-of-x-29-in-expression-1-x-5-x-7-x-9-29-

Question Number 84998 by jagoll last updated on 18/Mar/20

what is coefficient of x^{29}

in expression {(1 + x^{5} + x^{7} + x^{9})}^{29}

Answered by mind is power last updated on 18/Mar/20

29 = 5 a + 7 b + 9 c

a + b + c + d = 29

c = 1 \Rightarrow 20 = 5 a + 7 b \Rightarrow b = 0, a = 4

\Rightarrow d = 24

c = 0

29 = 5 a + 7 b \Rightarrow a = 3, b = 2,

d = 24

k = (a, b, c)

{(1 + x^{5} + x^{7} + x^{9})}^{29} = \sum_{a + b + c = 29 - d} \frac{29!}{a! . b! . c!} x^{5 a + 7 b + 9 c}

= (\frac{29!}{4! . 24!} + \frac{29!}{3! . 2! . 24!}) x^{29}

Commented by jagoll last updated on 18/Mar/20

thank you sir

Commented by jagoll last updated on 18/Mar/20

yes sir correct . via wolframalpha

Commented by mind is power last updated on 18/Mar/20

y e a h i f o r g e t 24!

Answered by mr W last updated on 18/Mar/20

{(1 + x^{5} + x^{7} + x^{9})}^{29}

= \sum_{a + b + c + d = 29} (_{a, b, c, d}^{29}) 1^{d} {(x^{5})}^{a} {(x^{7})}^{b} {(x^{9})}^{c}

= \sum_{a + b + c + d = 29} (_{a, b, c, d}^{29}) x^{5 a + 7 b + 9 c}

w i t h (_{a, b, c, d}^{29}) = \frac{29!}{a! b! c! d!}

e x a m p l e : t e r m x^{29} :

5 a + 7 b + 9 c = 29

\Rightarrow a = 3, b = 2, c = 0 \Rightarrow d = 24

\Rightarrow a = 4, b = 0, c = 1 \Rightarrow d = 24

t h e c o e f f i c i e n t o f x^{29} :

\frac{29!}{3! 2! 24!} + \frac{29!}{4! 24!} = 29 \times 28 \times 27 \times 26 \times 25 (\frac{1}{12} + \frac{1}{24})

= 1781325

e x a m p l e : t e r m x^{50} :

5 a + 7 b + 9 c = 50

\Rightarrow a = 3, b = 5, c = 0 \Rightarrow d = 21

\Rightarrow a = 10, b = 0, c = 0 \Rightarrow d = 19

\Rightarrow a = 1, b = 0, c = 5 \Rightarrow d = 23

t h e c o e f f i c i e n t o f x^{50} :

\frac{29!}{3! 5! 21!} + \frac{29!}{10! 19!} + \frac{29!}{5! 23!} = 263 240 250

Commented by mr W last updated on 18/Mar/20

Commented by mr W last updated on 18/Mar/20

m y r e s u l t c o i n c i d e s w i t h t h a t f r o m

w o l f r a m a l p h a, s e e b e l o w .

Commented by jagoll last updated on 19/Mar/20

sir w . thank you . i^{'} m wrong

operate my wolframaplha .

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