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Question Number 55100 by aseerimad last updated on 17/Feb/19

Commented by aseerimad last updated on 17/Feb/19
how no.14 is done?.....pls help
Commented by aseerimad last updated on 17/Feb/19
thank you sir. But can u please convert the answer to the form that I gave?
Commented by aseerimad last updated on 17/Feb/19

$t h e a n s w e r i s 2 (r - 1) \frac{(n - 2)!}{(n - r)!} b u t h o w ?$
Commented by mr W last updated on 17/Feb/19

$p l e a s e c h e c k t h e q u e s t i o n o r t h e a n s w e r .$ $2 (r - 1) \frac{(n - 2)!}{(n - r)!} i s t h e n u m b e r o f a r r a n g e m e n t s$ $t h a t t w o t h i n g s r e m a i n t o g e t h e r .$
Commented by aseerimad last updated on 17/Feb/19
sorry. actually these are what I got. May be there's a problem with the answer. Thanks anyway for your efforts!
	Answered by mr W last updated on 17/Feb/19

	$w e t a k e r t h i n g s f r o m n t h i n g s w i t h r ⩾ 2 .$ $t h e r e a r e t o t a l l y P_{r}^{n} w a y s .$ $n u m b e r o f w a y s s u c h t h a t t w o t h i n g s$ $(s a y A a n d B) a r e t o g e t h e r :$ $w e t a k e t h e r e s t r - 2 t h i n g s f r o m t h e$ $r e s t n - 2 t h i n g s, t h e r e a r e C_{r - 2}^{n - 2} w a y s t o$ $d o t h a t . s i n c e A a n d B s h o u l d b e t o g e t h e r,$ $a s A B o r a s B A, w e a r r a n g e i n f a c t$ $r - 1 t h i n g s, i . e . A B a n d t h e r e s t r - 2$ $t h i n g s, t h e r e a r e 2 \times (r - 1)! a r r a n g e m e n t s,$ $t o t a l l y t h e r e a r e C_{r - 2}^{n - 2} \times 2 \times (r - 1)! w a y s$ $t h a t t h e t w o t h i n g s a r e t o g e t h e r .$ $t h e r e f o r e n u m b e r o f w a y s s u c h t h a t$ $t w o t h i n g s a r e n o t t o g e t h e r i s$ $P_{r}^{n} - C_{r - 2}^{n - 2} \times 2 \times (r - 1)!$ $= \frac{n!}{(n - r)!} - \frac{(n - 2)! \times 2 \times (r - 1)!}{(r - 2)! (n - r)!}$ $= \frac{n (n - 1) (n - 2)!}{(n - r)!} - \frac{(n - 2)! \times 2 \times (r - 1)}{(n - r)!}$ $= \frac{(n - 2)!}{(n - r)!} [n (n - 1) - 2 (r - 1)]$
	Commented by aseerimad last updated on 17/Feb/19
	may the Almighty bless you with goodness!
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