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


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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

	${(1 + x + x^{2} + x^{3} + . . .)}^{3}$ $= \frac{1}{{(1 - x)}^{3}} = \underset{k = 0}{\sum^{\infty}} C_{2}^{k + 2} x^{k}$ $k = 18 :$ $C_{2}^{20} = \frac{20 \times 19}{2} = 190$
	Answered by mr W last updated on 07/Oct/20

	$x + y = n$ $x = 0, 1, . . ., n \Rightarrow n + 1 s o l u t i o n s$ $n = 18 - z = 0 . . 18$ $\Rightarrow t o t a l l y 1 + 2 + . . . + 19 = \frac{19 \times 20}{2} = 190$
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