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Question Number 53043 by Tinkutara last updated on 16/Jan/19

Commented by tanmay.chaudhury50@gmail.com last updated on 17/Jan/19

$e x c e l l e n t_{} q u e s t i o n . . .$
	Answered by mr W last updated on 17/Jan/19

	${l e t}^{'} s s a y t h e t w o n u m b e r s a r e X a n d Y$ $w i t h X < Y$ $f o r e a c h X i n 1 - 90 t h e r e a r e 10 Y s$ $\Rightarrow t o t a l l y 90 \times 10 = 900$ $f o r X = 91 : 9 Y s$ $f o r X = 92 : 8 Y s$ $f o r X = 93 : 7 Y s$ $. . .$ $f o r X = 99 : 1 Y$ $\Rightarrow 9 + 8 + . . . + 1 = 45$ $900 + 45 = 945$ $i . e . t h e r e a r e 945 w a y s .$ $(a) (b) (c) a r e a l l c o r r e c t .$
	Commented by Tinkutara last updated on 18/Jan/19
	Thank you Sir!
	Answered by tanmay.chaudhury50@gmail.com last updated on 18/Jan/19

	$d i f f e r e n c e = 1$ $(1, 2)$ $(2, 3) 99 c_{1}$ $. . .$ $. . . (99, 100)$ $d i f = 2$ $(1, 3)$ $2, 4$ $3, 5$ $. . . 98 c_{1}$ $. . .$ $98, 100$ $d i f = 3$ $(1, 4)$ $(2, 5)$ $(3, 6)$ $. . . 97 c_{1}$ $. . .$ $(97, 100)$ $s o a n s w e r i s$ $99 c_{1} + 98 c_{1} + 97 c_{1} + 96 c_{1} + 95 c_{1} + 94 c_{1} + 93 c_{1} + 92 c_{1}$ $+ 91 c_{1} + 90 c_{1}$ $= 99 + 98 + 97 + . . . + 90$ $= \frac{10}{2} [2 \times 99 + (10 - 1) \times - 1]$ $= 5 (198 - 9) = 5 \times 189 = 945$
	Commented by Tinkutara last updated on 18/Jan/19
	Thank you Sir!
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