in how many ways can the number n be written as a sum of three positive integers if representations differing in the order of the terms are considered to be different?


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Question Number 153220 by talminator2856791 last updated on 05/Sep/21

$in how many ways can the number$ $n be written as a sum of three positive$ $integers if representations differing$ $in the order of the terms are considered$ $to be different ?$
	Answered by mr W last updated on 05/Sep/21

	$a + b + c = n$ $a ⩾ 1, b ⩾ 1, c ⩾ 1$ ${(x + x^{2} + x^{3} + . . .)}^{3} = \frac{x^{3}}{{(1 - x)}^{3}} = \underset{k = 0}{\sum^{\infty}} C_{2}^{k + 2} x^{k + 3}$ $k + 3 = n \Rightarrow k + 1 = n - 1$ $c o e f . o f x^{n} i s C_{2}^{n - 1}, t h a t m e a n s t h e r e$ $a r e C_{2}^{n - 1} w a y s .$ $e x a m p l e n = 5 :$ $C_{2}^{4} = 6 w a y s$ $5 = 1 + 1 + 4 = 1 + 4 + 1 = 1 + 2 + 2 = 2 + 1 + 2$ $= 2 + 2 + 1 = 4 + 1 + 1$
	Commented by Tawa11 last updated on 06/Sep/21

	$great sir .$
	Commented by mr W last updated on 06/Sep/21

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