Find-the-coefficient-of-x-9-from-expression-1-x-1-2x-2-1-3x-3-1-4x-4-1-5x-5-1-10x-10-

Question Number 151211 by EDWIN88 last updated on 19/Aug/21

F i n d t h e c o e f f i c i e n t o f x^{9}

f r o m e x p r e s s i o n

(1 + x) (1 + 2 x^{2}) (1 + 3 x^{3}) (1 + 4 x^{4}) (1 + 5 x^{5}) \dots (1 + 10 x^{10})

Answered by Olaf_Thorendsen last updated on 19/Aug/21

f (x) = \underset{n = 1}{\prod^{10}} (1 + {n x}^{n})

∙ 9 = 9 + 0 : coeff = 9 \times 1 = 9

∙ 9 = 8 + 1 : coeff = 8 \times 1 = 8

∙ 9 = 7 + 2 : coeff = 7 \times 2 = 14

∙ 9 = 6 + 3 = 6 + 2 + 1 :

coeff = 6 \times 3 + 6 \times 2 \times 1 = 30

∙ 9 = 5 + 4 = 5 + 3 + 1

coeff = 5 \times 4 + 5 \times 3 \times 1 = 35

∙ 9 = 4 + 3 + 2

coeff = 4 \times 3 \times 2 = 24

coeff of x^{9} = 9 + 8 + 14 + 30 + 35 + 24 = 120

Commented by liberty last updated on 19/Aug/21

yes typo in part ∙ 9 = 7 + 2

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