A delegation of 4 students is to be selected from a total of 12 students. In how many ways can the delegation be selected if: 1) 2 particular students wish to be included together only in the delegation? 2) 2 particular students refuse to be together and 2 other particular students wish to be together only in the delegation?


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Question Number 50258 by rahul 19 last updated on 15/Dec/18

$A d e l e g a t i o n o f 4 s t u d e n t s i s t o b e$ $s e l e c t e d f r o m a t o t a l o f 12 s t u d e n t s .$ $I n h o w m a n y w a y s c a n t h e d e l e g a t i o n$ $b e s e l e c t e d i f :$ $1) 2 p a r t i c u l a r s t u d e n t s w i s h t o b e$ $i n c l u d e d t o g e t h e r o n l y i n t h e d e l e g a t i o n ?$ $2) 2 p a r t i c u l a r s t u d e n t s r e f u s e t o b e$ $t o g e t h e r a n d 2 o t h e r p a r t i c u l a r s t u d e n t s$ $w i s h t o b e t o g e t h e r o n l y i n t h e d e l e g a t i o n ?$
	Answered by mr W last updated on 15/Dec/18

	$A, B a r e t h e t w o s t u d e n t s w h o w a n t$ $t o b e t o g e t h e r i n t h e d e l e g a t i o n .$ $C, D a r e t h e t w o s t u d e n t s w h o r e f u s e$ $t o b e t o g e t h e r i n t h e d e l e g a t i o n .$ $1)$ $w i t h A a n d B : C_{2}^{10} w a y s$ $w i t h o u t A a n d B : C_{4}^{10} w a y s$ $\Rightarrow C_{2}^{10} + C_{4}^{10} = 255 w a y s$ $2)$ $w i t h A a n d B : C_{2}^{10} - 1$ $t o s e l e c t t h e o t h e r t w o d e l e g a t e s f r o m$ $t h e o t h e r 10 s t u d e n t s, t h e r e a r e C_{2}^{10}$ $w a y s, b u t t h e s e l e c t i o n w i t h C a n d D$ $t o g e t h e r m u s t b e e x c l u d e d, t h e r e f o r e$ $t h e r e a r e o n l y C_{2}^{10} - 1 w a y s .$ $w i t h o u t A a n d B : C_{4}^{10} - C_{2}^{8}$ $a l l 4 d e l e g a t e s w i l l b e s e l e c t e d f r o m t h e$ $o t h e r 10 s t u d e n t s, t h e r e a r e C_{4}^{10} w a y s,$ $w i t h C a n d D t o g e t h e r t h e r e a r e C_{2}^{8}$ $w a y s w h i c h m u s t b e e x c l u d e d, t h e r e f o r e$ $t h e r e a r e o n l y C_{4}^{10} - C_{2}^{8} w a y s .$ $\Rightarrow C_{2}^{10} - 1 + C_{4}^{10} - C_{2}^{8} = 255 - 29 = 226 w a y s$
	Commented by rahul 19 last updated on 15/Dec/18
	thank you sir!��
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