How many 3-digits positive integers are there such the sum of the digits is 10?


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Question Number 123681 by ZiYangLee last updated on 27/Nov/20

$How many 3 - digits positive integers$ $are there such the sum of the digits$ $is 10 ?$
	Answered by mr W last updated on 27/Nov/20

	$a b c w i t h$ $a + b + c = 10$ $a = 1 . . 9$ $b, c = 0 . . 9$ $(x + x^{2} + x^{3} + . . . + x^{9}) {(1 + x + x^{2} + . . . + x^{9})}^{2}$ $= \frac{x (1 - x^{9}) {(1 - x^{10})}^{2}}{{(1 - x)}^{3}}$ $= . . . + 54 x^{10} + . . .$ $\Rightarrow t h e r e a r e 54 s u c h 3 d i g i t n u m b e r s .$
	Commented by malwan last updated on 27/Nov/20

	$\frac{(x - x^{10}) (1 - 2 x^{10} + x^{20})}{1 - 3 x + 3 x^{2} - x^{3}}$ $= \frac{(x - x^{10} - 2 x^{11} + 2 x^{20} + x^{21} - x^{30})}{1 - 3 x + 3 x^{2} - x^{3}}$ $t h e n d i v i d e$ $y o u w i l l g e t . . . x^{10}$ $i s i t r i g h t s i r ?$
	Commented by mr W last updated on 27/Nov/20

	$i d i d i n t h i s w a y :$ $\frac{1}{{(1 - x)}^{3}} = \underset{k = 0}{\sum^{\infty}} C_{2}^{k + 2} x^{k}$ $\frac{x (1 - x^{9}) {(1 - x^{10})}^{2}}{{(1 - x)}^{3}}$ $= x (1 - x^{9}) {(1 - x^{10})}^{2} \underset{k = 0}{\sum^{\infty}} C_{2}^{k + 2} x^{k}$ $c o e f . o f x^{10} :$ $C_{2}^{9 + 2} - C_{2}^{0 + 2} = C_{2}^{11} - C_{2}^{2} = \frac{11 \times 10}{2} - 1 = 54$
	Commented by malwan last updated on 28/Nov/20

	$f a n t a s t i c s i r$ $t h a n k y o u$
	Answered by floor(10²Eta[1]) last updated on 27/Nov/20

	$109$ $118$ $127$ $136$ $145$ $154$ $163$ $172$ $181$ $190$ $(10 cases)$ $208$ $217$ $226$ $235$ $244$ $253$ $262$ $271$ $280$ $(9 cases)$ $307$ $316$ $325$ $334$ $343$ $352$ $361$ $370$ $(8 cases)$ $. . .$ $901$ $910$ $(2 cases)$ $total cases : 10 + 9 + 8 + . . . + 2 = 54$
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