in how many ways can you distribute 20 different books to 5 students such that each student gets at least 2 books?


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Question Number 179818 by mr W last updated on 02/Nov/22

$i n h o w m a n y w a y s c a n y o u d i s t r i b u t e$ $20 d i f f e r e n t b o o k s t o 5 s t u d e n t s s u c h$ $t h a t e a c h s t u d e n t g e t s a t l e a s t 2 b o o k s ?$
	Answered by a.lgnaoui last updated on 03/Nov/22

	$3 4 4 4 5 4 3 4 4 5$ $3 4 4 5 4 4 3 4 5 4$ $3 4 5 4 4 4 3 5 4 4$ $3 5 4 4 4 5 3 4 4 4$ $4 4 3 4 5 4 4 4 3 5$ $4 4 3 5 4 4 4 5 3 4$ $4 5 3 4 4 4 5 4 3 4$ $5 4 3 4 4 5 4 4 3 4$ $4 4 4 5 3$ $4 4 5 4 3 t o t a l = 20 w a y s$ $4 5 4 4 3$ $5 4 4 4 3$
	Commented by a.lgnaoui last updated on 03/Nov/22

	$i n m y a n s w e r I c o n s i d e r e d$ $n o d i f f e r e n c e b e t w e n b o o k s$ $o r d e r 1 2 3 4 5 (f o r s u d e n t s)$ $3, 4, 5 N u m b e r o f b o o k s f o r$ $e a c h s t u d e n t .$
	Commented by mr W last updated on 03/Nov/22

	$a t l e a s t 2 b o o k s \neq m o r e t h a n 2 b o o k s$ $b e s i d e s t h e b o o k s a r e d i f f e r e n t a n d$ $t h e s t u d e n t s a r e a l s o d i f f e r e n t$ $(l o g i c a l l y!)$
	Answered by Acem last updated on 04/Nov/22

	$P r i m i t i v e S o l u t i o n d r a f t$ $A) d i s t r i b u t i n g 2 d i f . b k s . t o e a c h s t u .$ $A = C_{2}^{20} C_{2}^{18} C_{2}^{16} C_{2}^{14} C_{2}^{12} 5! = 2 514 159 648 000$ $B) d i s t r i b u t i n g t h e o t h e r 10 d i f . b k s . t o 5 s t u .$ $B = C_{10}^{10} \times 5 + C_{9}^{10} \times 5 . C_{1}^{1} \times 4 + C_{8}^{10} \times 5 C_{2}^{2} \times 4$ $. . . . + C_{2}^{10} \times 5 C_{8}^{8} \times 4 + C_{1}^{10} \times 5 C_{9}^{9} \times 4$ $N u m W a y s = A + B$
	Answered by mr W last updated on 03/Nov/22

	$t h e n u m b e r o f w a y s t o d i s t r i b u t e 20$ $b o o k s t o 5 s t u d e n t s s u c h t h a t e a c h$ $s t u d e n t g e t s a t l e a s t 2 b o o k s i s t h e$ $c o e f . o f x^{20} i n t h e e x p a n s i o n o f$ $20! {(e^{x} - 1 - x)}^{5} w h i c h i s$ $64 341 775 001 400 .$
	Commented by mr W last updated on 03/Nov/22

	Commented by mr W last updated on 03/Nov/22

	$i s o l v e d i n a w a y w h i c h s e e m s t o b e a$ $g o o d w a y t o m e . i {c a n}^{'} t a n d {w o n}^{'} t$ $a n s w e r w h y i d i d i t d i f f e r e n t l y a s$ $y o u .$ $m y t h o u g h t w a s : i n s t e p 1 i l e t e a c h$ $s t u d e n t g e t o n e b o o k a n d i n s t e p 2$ $e a c h g e t s a t l e a s t o n e b o o k .$ $i c o n s i d e r e d a g a i n a n d f o u n d o u t t h a t$ $i c a n a l s o s o l v e t h e q u e s t i o n w i t h m y$ $m e t h o d (u s i n g g e n e r a t i n g f u n c t i o n)$ $e v e n w i t h o u t s t e p 1 . s o i h a v e m o d i f i e d$ $m y s o l u t i o n c o r r e s p o n d i n g l y .$ $i {c o u l d n}^{'} t t o t a l l y f o l l o w y o u r s o l u t i o n,$ $b u t i t h i n k i t i s n o t c o r r e c t . y o u c a n$ $c h e c k y o u r s o l u t i o n f o r e x a m p l e t h a t$ $w e d i s t r i b u t e 5 b o o k s t o 2 s t u d e n t s$ $a n d e a c h g e t s a t l e a s t 2 b o o k s . t h e$ $c o r r e c t a n s w e r s h o u l d b e 20 . b u t y o u r$ $m e t h o d g e t s a l r e a d y i n s t e p 1$ $A = C_{2}^{5} C_{2}^{3} 2! = 60 . t h a t m e a n s t h e$ $m e t h o d i s w r o n g . s o p l e a s e r e c h e c k$ $y o u r o w n s o l u t i o n .$
	Commented by Acem last updated on 03/Nov/22

	$H e l l o,$ $F o r t h e 1 s t s t e p, w h y o n l y 5 b o o k s n o t 10$ $w e c a n c h o o s e r a n d o m l y 2 b o o k s a n d g i v e$ $t h e m t o s o m e o n e a s f o l l o w$ $A) d i s t r i b u t i n g 2 d i f . b k s . t o e a c h s t u .$ $A = C_{2}^{20} C_{2}^{18} C_{2}^{16} C_{2}^{14} C_{2}^{12} 5! = 2 514 159 648 000$
	Commented by Acem last updated on 03/Nov/22

	$T h a n k y o u m y f r i e n d f o r e v e r y t h i n g, i n f a c t i a m$ $s u r e t h a t i^{'} v e s o m e t h i n g m i s s i n g w i t h t h e 2 s t e p$ $t h a t i s w h y {d i d n}^{'} t c o m p t i t r e s u l t, a n d w i l l t r y$ $t o f i x i t l a t e r, b u t n o w a m s h o c k e d a b o u t t h e 1 s t$ $s t e p, o k i w i l l r e c h e c k i t w h e n f i n i s h m y w o r k .$ $H a v e a n i c e t i m e$
	Commented by Acem last updated on 03/Nov/22

	$H e l l o a g a i n,$ $I g o t t h e m i s t a k e$ $Prime causes double exponent: use braces to clarify$ $a n d t h e t r u e i s :$ $A = C_{2}^{20} C_{2}^{18} C_{2}^{16} C_{2}^{14} C_{12}^{12} 5! = 38 093 328 000 ✓$ $N o w, d e a l i n g w i t h 12, 11, 10, . . . b o o k s i s e a s i e r$ $t h a n d e a l i n g w i t h 15, 14, . . . b o o k s$ $A n y w a y i g o t t h e r e s u l t : 62 174 055 528 000$ $I w i l l f e e d b a c k m y m e t h o d v e r y s o o n$ $I f y o u r s i s t r u e t h a t m e a n i s t i l l m i s s i n g s o m e$ $c a s e s o f d i s t r u b t i o n$ $P . s : p l e a s e m a k e t h e q u e s t i o n w i t h l e s s n u m b e r$ $a s 9 b o o k s f o r e x a m p l e a s l o n g a s t h e i d e a o f$ $t h e s o l u t i o n i s w e l l k n o w n$ $T h a n k y o u$
	Commented by mr W last updated on 03/Nov/22

	$a s i h a v e s a i d, y o u c a n c h e c k y o u r$ $m e t h o d w i t h t h e m o s t s i m p l e c a s e :$ $5 b o o k s t o 2 s t u d e n t, a t l e a s t 2 b o o k s$ $f o r e a c h s t u d e n t .$ $i g o t t h e a n s w e r : 20 w a y s . t h i s i s$ $t r u e a n d c a n b e e a s i l y p r o v e d .$
	Commented by Acem last updated on 03/Nov/22

	$I k n o w t h a t, b u t a s t h e n u m b e r o f i t e m s i n c r e a s e s$ $t h e b r a n c h e s i n c r e a s e, a n d h e r e i f w e n e g l e c t$ $o n e c a s e o f d i s t r u b t i o n w e w o u l d l o s e i t s b r a n c h e s$
	Commented by mr W last updated on 03/Nov/22

	$t h e n y o u c a n p u t a n e x a m p l e b y$ $y o u r s e l f w h i c h y o u t h i n k i s g o o d .$
	Answered by Acem last updated on 04/Nov/22

	$T h e s o l v e i n o n e s t a g e$ $N u m W a y s =$ $5! [C_{12}^{20} C_{2}^{8} C_{2}^{6} C_{2}^{4} C_{2}^{2} + C_{11}^{20} C_{3}^{9} C_{2}^{6} C_{2}^{4} C_{2}^{2} +$ $C_{10}^{20} (C_{4}^{10} C_{2}^{6} C_{2}^{4} C_{2}^{2} + C_{3}^{10} C_{3}^{7} C_{2}^{4} C_{2}^{2} +$ $C_{9}^{20} (C_{5}^{11} C_{2}^{6} C_{2}^{4} C_{2}^{2} + C_{4}^{11} C_{3}^{7} C_{2}^{4} C_{2}^{2} +$ $+ C_{3}^{11} C_{3}^{8} C_{3}^{5} C_{2}^{2}) +$ $C_{8}^{20} (C_{6}^{12} C_{2}^{6} C_{2}^{4} C_{2}^{2} + C_{5}^{12} C_{3}^{7} C_{2}^{4} C_{2}^{2} +$ $C_{4}^{12} C_{4}^{8} C_{2}^{4} C_{2}^{2} + C_{4}^{12} C_{3}^{8} C_{3}^{5} C_{2}^{2} +$ $C_{3}^{12} C_{3}^{9} C_{3}^{6} C_{3}^{3}) +$ $C_{7}^{20} (C_{7}^{13} C_{2}^{6} C_{2}^{4} C_{2}^{2} + C_{6}^{13} C_{3}^{7} C_{2}^{4} C_{2}^{2} +$ $C_{5}^{13} C_{4}^{8} C_{2}^{4} C_{2}^{2} + C_{5}^{13} C_{3}^{8} C_{3}^{5} C_{2}^{2}$ $C_{4}^{13} C_{4}^{9} C_{3}^{5} C_{2}^{2} + C_{4}^{13} C_{3}^{9} C_{3}^{6} C_{3}^{3})]$ $N u m W a y s = 62 174 055 528 000$ $T h i n g s a r e v e r y c l e a r i n f r o n t o f y o u$ $a n d s o f a r i h a v e n o t f e l t t h a t i^{'} v e o v e r l o o k e d$ $o n e o f t h e c a s e s$
	Commented by Acem last updated on 04/Nov/22

	$A) C_{3}^{5} C_{2}^{2} 2! = 20$ $B) (C_{4}^{6} C_{2}^{2} + C_{3}^{6} C_{3}^{3}) 2! = 70 n o t 50$ $C) O u r m a i n q u e s t i o n ‘ ‘ 20 b o o k s t o 5 s t u .^{″}$ $C a n y o u n o t e y o u r m e t h o d t o c o m p a r e o u r$ $d i s t r i b u t i o n m e t h o d s w i t h o u t u s i n g$ $s p e c i a l t a b l e s$
	Commented by mr W last updated on 04/Nov/22

	$i f y o u a r e n o t c o n v i n c e d t h a t y o u r$ $r e s u l t i s w r o n g, w h y n o t v e r i f y y o u r$ $s o l u t i o n w i t h s o m e s i m p l e c a s e s$ $w h i c h c a n b e e a s i l y c h e c k e d$ $‘ ‘ {m a n u a l l l y}^{″} ? f o r e x a m p l e$ $e x a m p l e 1 : 5 b o o k s t o 2 s t u d e n t s,$ $a n s w e r s h o u l d b e 20 .$ $e x a m p l e 2 : 6 b o o k s t o 2 s t u d e n t s,$ $a n s w e r s h o u l d b e 50 .$
	Commented by mr W last updated on 04/Nov/22

	$A) 5 b o o k s t o 2 s t u d e n t s :$ $a n s w e r i s 20 . t h i s c a n b e c h e c k e d :$ $C_{2}^{5} \times 2 = 20$ $i c a n g e t t h e s a m e r e s u l t v i a c o e f . o f$ $x^{5} i n e x p a n s i o n o f 5! {(e^{x} - 1 - x)}^{2} .$ $B) 6 b o o k s t o 2 s t u d e n t s :$ $a n s w e r i s 50 . t h i s c a n b e c h e c k e d :$ $C_{2}^{6} \times 2 + C_{3}^{6} = 15 \times 2 + 20 = 50, n o t 70!$ $i n f a c t o n e c a n e n u m e r a t e a l l t h e s e$ $50 c a s e s e a s i l y .$ $i c a n g e t t h e s a m e r e s u l t v i a c o e f . o f$ $x^{6} i n e x p a n s i o n o f 6! {(e^{x} - 1 - x)}^{2} .$ $C) 20 b o o k s t o 5 s t u d e n t s :$ $t h e r e s u l t i s t h e c o e f . o f x^{20} i n t h e$ $e x p a n s i o n o f 20! {(e^{x} - 1 - x)}^{5}, w h i c h$ $i s 64 341 775 001 400 .$ $i c a n g e t t h i s r e s u l t ‘ ‘ {m a n u a l l y}^{″} i n a n$ $o t h e r w a y a s f o l l o w i n g .$ $t o d i s t r i b u t e 20 b o o k s i n t o 5 g r o u p s$ $s u c h t h a t e a c h g r o u p g e t s a t l e a s t$ $2 b o o k s, t h e r e a r e f o l l o w i n g 30$ $p o s s i b i l i t i e s :$ $12 + 2 + 2 + 2 + 2 \Rightarrow \frac{20!}{12! {(2!)}^{4} 4!} \times 5!$ $11 + 3 + 2 + 2 + 2 \Rightarrow \frac{20!}{11! 3! {(2!)}^{3} 3!} \times 5!$ $10 + 4 + 2 + 2 + 2 \Rightarrow \frac{20!}{10! 4! {(2!)}^{3} 3!} \times 5!$ $10 + 3 + 3 + 2 + 2 \Rightarrow \frac{20!}{10! {(3! 2!)}^{2} 2! 2!} \times 5!$ $9 + 5 + 2 + 2 + 2 \Rightarrow \frac{20!}{9! 5! {(2!)}^{3} 3!} \times 5!$ $9 + 4 + 3 + 2 + 2 \Rightarrow \frac{20!}{9! 4! 3! {(2!)}^{2} 2!} \times 5!$ $9 + 3 + 3 + 3 + 2 \Rightarrow \frac{20!}{9! {(3!)}^{3} 2! 3!} \times 5!$ $8 + 6 + 2 + 2 + 2 \Rightarrow \frac{20!}{8! 6! {(2!)}^{3} 3!} \times 5!$ $8 + 5 + 3 + 2 + 2 \Rightarrow \frac{20!}{8! 5! 3! {(2!)}^{2} 2!} \times 5!$ $8 + 4 + 4 + 2 + 2 \Rightarrow \frac{20!}{8! {(4! 2!)}^{2} 2! 2!} \times 5!$ $8 + 4 + 3 + 3 + 2 \Rightarrow \frac{20!}{8! 4! 2! {(3!)}^{2} 2!} \times 5!$ $8 + 3 + 3 + 3 + 3 \Rightarrow \frac{20!}{8! {(3!)}^{4} 4!} \times 5!$ $7 + 7 + 2 + 2 + 2 \Rightarrow \frac{20!}{{(7!)}^{2} {(2!)}^{3} 2! 3!} \times 5!$ $7 + 6 + 3 + 2 + 2 \Rightarrow \frac{20!}{7! 6! 3! {(2!)}^{2} 2!} \times 5!$ $7 + 5 + 4 + 2 + 2 \Rightarrow \frac{20!}{7! 5! 4! {(2!)}^{2} 2!} \times 5!$ $7 + 5 + 3 + 3 + 2 \Rightarrow \frac{20!}{7! 5! 2! {(3!)}^{2} 2!} \times 5!$ $7 + 4 + 4 + 3 + 2 \Rightarrow \frac{20!}{7! 3! 2! {(4!)}^{2} 2!} \times 5!$ $7 + 4 + 3 + 3 + 3 \Rightarrow \frac{20!}{7! 4! {(3!)}^{3} 3!} \times 5!$ $6 + 6 + 4 + 2 + 2 \Rightarrow \frac{20!}{{(6!)}^{2} 4! {(2!)}^{2} 2! 2!} \times 5!$ $6 + 6 + 3 + 3 + 2 \Rightarrow \frac{20!}{2! {(6!)}^{2} {(3!)}^{2} 2! 2!} \times 5!$ $6 + 5 + 5 + 2 + 2 \Rightarrow \frac{20!}{6! {(5!)}^{2} {(2!)}^{2} 2! 2!} \times 5!$ $6 + 5 + 4 + 3 + 2 \Rightarrow \frac{20!}{6! 5! 4! 3! 2!} \times 5!$ $6 + 5 + 3 + 3 + 3 \Rightarrow \frac{20!}{6! 5! {(3!)}^{3} 3!} \times 5!$ $6 + 4 + 4 + 4 + 2 \Rightarrow \frac{20!}{6! 2! {(4!)}^{3} 3!} \times 5!$ $6 + 4 + 4 + 3 + 3 \Rightarrow \frac{20!}{6! {(4!)}^{2} {(3!)}^{2} 2! 2!} \times 5!$ $5 + 5 + 5 + 3 + 2 \Rightarrow \frac{20!}{3! 2! {(5!)}^{3} 3!} \times 5!$ $5 + 5 + 4 + 4 + 2 \Rightarrow \frac{20!}{2! {(5!)}^{2} {(4!)}^{2} 2! 2!} \times 5!$ $5 + 5 + 4 + 3 + 3 \Rightarrow \frac{20!}{4! {(5!)}^{2} {(3!)}^{2} 2! 2!} \times 5!$ $5 + 4 + 4 + 4 + 3 \Rightarrow \frac{20!}{5! 3! {(4!)}^{3} 3!} \times 5!$ $4 + 4 + 4 + 4 + 4 \Rightarrow \frac{20!}{{(4!)}^{5} 5!} \times 5!$ $Σ : 64 341 775 001 400$
	Commented by mr W last updated on 04/Nov/22

	Commented by Acem last updated on 05/Nov/22

	$H e l l o a n d g o o d e v e n i n g m y b e s t f r i e n d$ $A t f i r s t a n d f o r t h e$ $C_{2}^{6} \times 2 + C_{3}^{6} = 15 \times 2 + 20 = 50, n o t 70!$ $y o u s a y . . . . . o r 3 b o o k s f o r e a c h s t u d e n t C_{3}^{6} C_{3}^{3}$ $y o u k n o w a s w e l l t h a t w e h a v e 2 w a y s f o r d i s t r u b t i n g$ $c a u s e w e h a v e d i f f e r e n t b o o k s t h a t i s w h y$ $i m u l t i p l y C_{3}^{6} b y 2 t o g e t f i n a l l y 70 w a y s$ $w a i t i n g f o r y o u r r e p l y S i r, a n d s u r e i w i l l$ $b e b a c k f o r t h e i m p o r t a n t i s s u e a b o u t m a i n q u s e t i o n$ $T h a n k s f o r y o u r e f f o r t s$
	Commented by mr W last updated on 05/Nov/22

	$a s y o u w r o t e C_{3}^{6} y o u h a v e a l r e a d y$ $c o u n t e d a l l c a s e s . e x a m p l e : s a y t h e$ $b o o k s a r e a, b, c, d, e, f . w h e n w e s e l e c t$ $b o o k s a, b, c f o r t h e f i r s t p e r s o n, t h e$ $s e c o n d p e r s o n g e t s a u t o m a c t i c a l l y$ $t h e b o o k s d, e, f . w h e n t h e f i r s t p e r s o n$ $g e t s d, e, f, t h e s e c o n d p e r s o n s g e t s$ $a u t o m a t i c a l l y a, b, c . t h a t m e a n s b o t h$ $c a s e s a r e c o m p l e t e l y i n c l u d e d i n C_{3}^{6} .$ $t h i s i s b e c a u s e b o t h p e r s o n s g e t$ $e q u a l n u m b e r o f b o o k s . {i t}^{'} s d i f f e r e n t$ $w h e n a p e r s o n g e t s 2 b o o k s a n d t h e$ $o t h e r p e r s o n 4 b o o k s, t h e n w e m u s t$ $w r i t e 2 \times C_{2}^{6} . t h i n k a b o u t i t, s h o u l d b e$ $c l e a r .$ $e x a m p l e 4 b o o k s f o r 2 p e r s o n s :$ $a + b c d o r b c d + a ✓$ $b + a c d o r a c d + b ✓$ $c + a b d o r a b d + c ✓$ $d + a b c o r a b c + d ✓$ $a b + c d o r c d + a b (?!)$ $a c + b d o r b d + a c (?!)$ $a d + b c o r b c + a d (?!)$ $b c + a d o r a d + b c (?!)$ $b d + a c o r a c + b d (?!)$ $c d + a b o r a b + c d (?!)$ $y o u c a n s e e w e c a n n o t m u l t i p l y b y 2$ $f o r t h e c a s e s w h e n b o t h p e o p l e g e t t h e$ $s a m e n u m b e r o f b o o k s .$
	Commented by Acem last updated on 05/Nov/22
	$Yes i agree with you, and with this in mind, let′s ask the following essential question: How does the whole process work? A) Mr. w please come to get 3 books, “Books selection process C_( 3) ^( 6) ”here they are. the rest books for Acem...... it′s ok! B) “Books selection process C_( 3) ^( 6) ” {Math, Fluid mechanics, Steam generators} Now me and you like these science fields, the rest books are about other things. and here, i see that there′s still a different, ′in fact i want these books for me not you′ so, don′t we have different ways to distribute? with diferent books and people$
	$Y e s i a g r e e w i t h y o u, a n d w i t h t h i s i n m i n d, {l e t}^{'} s$ $a s k t h e f o l l o w i n g e s s e n t i a l q u e s t i o n :$ $H o w d o e s t h e w h o l e p r o c e s s w o r k ?$ $A) M r . w p l e a s e c o m e t o g e t 3 b o o k s,$ $‘ ‘ B o o k s s e l e c t i o n p r o c e s s C_{3}^{6}^{″} h e r e t h e y a r e .$ $t h e r e s t b o o k s f o r A c e m . . . . . . {i t}^{'} s o k!$ $B) ‘ ‘ B o o k s s e l e c t i o n p r o c e s s C_{3}^{6}^{″}$ ${M a t h, F l u i d m e c h a n i c s, S t e a m g e n e r a t o r s}$ $N o w m e a n d y o u l i k e t h e s e s c i e n c e f i e l d s, t h e r e s t$ $b o o k s a r e a b o u t o t h e r t h i n g s .$ $a n d h e r e, i s e e t h a t {t h e r e}^{'} s s t i l l a d i f f e r e n t,$ $^{'} i n f a c t i w a n t t h e s e b o o k s f o r m e n o t {y o u}^{'}$ $s o, {d o n}^{'} t w e h a v e d i f f e r e n t w a y s t o d i s t r i b u t e ?$ $w i t h d i f e r e n t b o o k s a n d p e o p l e$
	Commented by mr W last updated on 05/Nov/22

	$i {c a n}^{'} t f o l l o w y o u s i r .$ $t h e r e a r e 50 w a y s t o d i s t r i b u t e 6$ $d i f f e r e n t b o o k s t o 2 s t u d e n t s s u c h$ $t h a t e a c h g e t s a t l e a s t 2 b o o k s .$ $d o y o u a g r e e o r n o t ?$ $i f y o u a g r e e, {t h a t}^{'} s f i n e, w e c a n c l o s e$ $t h e c a s e .$ $i f y o u {d o n}^{'} t a g r e e, y o u a r e w r o n g, a n d$ $i a l s o c l o s e t h e c a s e, b e c a u s e i {c a n}^{'} t$ $c o n t r i b u t e a n y m o r e . y o u o n l y n e e d t o$ $e n u m e r a t e a l l p o s s i b i l i t i e s a n d t h e n$ ${y o u}^{'} l l s e e t h a t t h e r e a r e i n d e e d o n l y$ $50 d i f f e r e n t w a y s .$
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