Q-what-is-the-greatest-coefficient-of-the-following-expansion-P-a-b-c-d-10-

Question Number 173531 by mnjuly1970 last updated on 13/Jul/22

Q :

w h a t i s t h e g r e a t e s t c o e f f i c i e n t

o f t h e f o l l o w i n g e x p a n s i o n .

P := {(a + b + c + d)}^{10}

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Answered by mr W last updated on 13/Jul/22

c o e f f i c i e n t = \frac{10!}{m! n! h! k!}

w i t h 0 ⩽ m, n, h, k ⩽ 10 a n d m + n + h + k = 10

d u e t o s y m m e t r y, l a r g e s t c o e f f i c i e n t

i s w h e n m, n, h, k a r e a t m o s t e q u a l t o

e a c h o t h e r . t w o c a s e s s h o u l d e b e c h e c k e d :

m = n = h = 3, k = 1 o r

m = n = 3, k = h = 2

{(3!)}^{3} \times 1! = 216

{(3!)}^{2} \times {(2!)}^{2} = 144 < 216

\Rightarrow l a r g e s t c o e f . = \frac{10!}{144} = 25200 ✓

Commented by mnjuly1970 last updated on 13/Jul/22

t h a n k s a l o t s i r W

Commented by Tawa11 last updated on 13/Jul/22

Great sir

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