The teacher can choose in 560 ways, provided that there are three students in each team. Knowing that five students do not want to participate, find the number of people willing to participate


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Question Number 201729 by hardmath last updated on 11/Dec/23

The teacher can choose in 560 ways, provided that there are three students in each team. Knowing that five students do not want to participate, find the number of people willing to participate
Commented by mr W last updated on 11/Dec/23

$y o u {d i d n}^{'} t s t a t e c l e a r l y e n o u g h,$ $w h a t t h e t e a c h e r d o e s ? w h a t d o e s$ $t h e t e a c h e r c h o o s e i n 560 w a y s ?$
Commented by mr W last updated on 11/Dec/23

$w h e n y o u j u s t w a n t t o f i n d h o w m a n y$ $p e o p l e a r e w i l l i n g t o p a r t i c i p a t e,$ $t h e n i t {d o e s n}^{'} t m a t t e r h o w m a n y$ $p e o p l e {d o n}^{'} t w a n t t o p a r t i c i p a t e .$
Commented by hardmath last updated on 11/Dec/23

$yes dear professor$
Commented by hardmath last updated on 11/Dec/23

$dear professor, Is there a solution ?$
Commented by mr W last updated on 11/Dec/23

$t h e f i r s t s e n t e n c e i s c u t o f f . i f y o u$ $e v e n {d o n}^{'} t k n o w w h a t i t i s, h o w c a n$ $i k n o w ?$
Commented by mr W last updated on 11/Dec/23

$w h e n {i t}^{'} s n o t c l e a r w h a t y o u r q u e s t i o n$ $i s, h o w i s a s o l u t i o n p o s s i b l e ?$
Commented by mr W last updated on 11/Dec/23

$i h a v e a s k e d y o u :$ $w h a t d o e s t h e t e a c h e r c h o o s e i n$ $560 w a y s ?$
Commented by mr W last updated on 11/Dec/23

$o r w h e n y o u r e a d y o u r q u e s t i o n, d o$ $y o u u n d e r s t a n d y o u r q u e s t i o n ?$
Commented by hardmath last updated on 11/Dec/23

Dear professor for example, what might be the right condition
Commented by hardmath last updated on 11/Dec/23

I don't know, dear professor, it was given on the condition that I asked for help
	Answered by esmaeil last updated on 11/Dec/23

	$(\begin{matrix} n - 5 \\ 3 \end{matrix}) = 56 o \to$ $\frac{(n - 5)!}{3! (n - 8)!} = 560 \to$ $(n - 5) (n - 6) (n - 7) = 3360$ $n = 21$
	Commented by mr W last updated on 11/Dec/23

	$w h a t i s t h e q u e s t i o n ?$ $21 w i l l i n g p e o p l e ? b u t 5 a m o n g t h e m$ $d o n o t w a n t t o p a r t i c i p a t e ? t h e n$ $t h e y a r e n o t w i l l i n g p e o p l e .$
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