there are 5 points on a line and 4 points on a parallel line. how many quadrilaterals can be formed with these points as vertrices?


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Question Number 180585 by mr W last updated on 14/Nov/22

$t h e r e a r e 5 p o i n t s o n a l i n e a n d 4$ $p o i n t s o n a p a r a l l e l l i n e .$ $h o w m a n y q u a d r i l a t e r a l s c a n b e$ $f o r m e d w i t h t h e s e p o i n t s a s$ $v e r t r i c e s ?$
	Answered by nikif99 last updated on 14/Nov/22

	$A n y t w o p o i n t s o u t o f 5 i n t h e 1 s t$ $l i n e = C_{2}^{5} = 10 c o m b i n e d w i t h a n y t w o$ $p o i n t s o u t o f 4 i n t h e 2 n d l i n e =$ $C_{2}^{4} = 6 m a k e a t o t a l o f 60 .$
	Commented by mr W last updated on 14/Nov/22

	$w i t h 4 p o i n t s w e c a n f o r m 3 d i f f e r e n t$ $q u a d r i l a t e r s, n o t o n l y o n e .$
	Commented by nikif99 last updated on 14/Nov/22

	$I f I w e l l u n d e r s t o o d y o u r c o m m e n t S i r,$ $y o u m e a n t h e r e a r e 3 p o s s i b l e$ $q u a d r i l a t e r s h a v i n g 2 v e r t i c e s i n$ $a n y 4 p o i n t s o f a l i n e ?$
	Commented by nikif99 last updated on 14/Nov/22

	Commented by mr W last updated on 14/Nov/22

	$i m e a n : w h e n w e h a v e 4 p o i n t s, w e c a n$ $f o r m 3 d i f f e r e n t q u a d r i l a t e r a l s,$ $s e e e x a m p l e b e l o w .$ $s o t a t a l l y 60 \times 3 = 180 q u a d r i l a t e r a l s$ $c a n b e f o r m e d .$
	Commented by mr W last updated on 14/Nov/22

	Commented by nikif99 last updated on 14/Nov/22

	$P E R F E C T . I s e e . Y o u a r e r i g h t, S i r .$
	Answered by cortano1 last updated on 14/Nov/22

	$(\begin{matrix} 9 \\ 4 \end{matrix}) - (\begin{matrix} 5 \\ 4 \end{matrix}) - (\begin{matrix} 4 \\ 4 \end{matrix})$ $= \frac{9.8.7.6}{4.3.2.1} - 5 - 1$ $= 126 - 6 = 120$
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