Find-the-number-m-of-non-negative-integer-solution-of-x-y-z-18-

Question Number 116881 by bemath last updated on 07/Oct/20

Find the number m of non negative

integer solution of x + y + z = 18

Answered by john santu last updated on 07/Oct/20

w e c a n v i e w e a c h s o l u t i o n, s a y x = 3, y = 7, z = 8

a s a c o m b i n a t i o n o f r = 18 o b j e c t s

c o n s i s t i n g o f 3 a^{'} s, 7 b^{'} s a n d 8 c^{'} s, w h e r e

t h e r e a r e M = 3 k i n d s o f o b j e c t s a^{'} s, b^{'} s

a n d c^{'} s . T h e n m = C (r + M - 1, M - 1)

= C (20, 2) = \frac{20 \times 19}{2 \times 1} = 190

Answered by mr W last updated on 07/Oct/20

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

= \frac{1}{{(1 - x)}^{3}} = \underset{k = 0}{\sum^{\infty}} C_{2}^{k + 2} x^{k}

k = 18 :

C_{2}^{20} = \frac{20 \times 19}{2} = 190

Answered by mr W last updated on 07/Oct/20

x + y = n

x = 0, 1, \dots, n \Rightarrow n + 1 s o l u t i o n s

n = 18 - z = 0 . . 18

\Rightarrow t o t a l l y 1 + 2 + \dots + 19 = \frac{19 \times 20}{2} = 190

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