in how many ways we can distribute 6 distinct balls in 3 identical boxes


Question and Answers Forum

All Questions Topic List
Algebra Questions
Previous in All Question Next in All Question
Previous in Algebra Next in Algebra
Question Number 212522 by Nadirhashim last updated on 16/Oct/24

$i n h o w m a n y w a y s w e$ $c a n d i s t r i b u t e 6 d i s t i n c t$ $b a l l s i n 3 i d e n t i c a l b o x e s$
	Answered by mehdee7396 last updated on 16/Oct/24

	$6 / 0 / 0 o r 5 / 1 / 0 o r 4 / 2 / 0 o r 4 / 1 / 1 o r 3 / 3 / 0 / o r 3 / 2 / 1 o r 2 / 2 / 2$ $C (6, 6) + C (5, 1) C (1, 1) + C (6, 4) C (2, 2) + C (6, 4) C (2, 1) C (1, 1) + C (6, 3) C (3, 3) + C (6, 3) C (3, 2) C (1, 1) + C (6, 2) C (4, 2) C (2, 2)$ $= 275 ✓$
	Commented by mr W last updated on 21/Oct/24

	${i t}^{'} s w r o n g s i r .$ $t h e n u m b e r o f w a y s t o d i s t r i b u t e n$ $d i s t i n c t o b j e c t s i n t o$ $n_{1} g r o u p s w i t h m_{1} o b j e c t s i n e a c h a n d$ $n_{2} g r o u p s w i t h m_{2} o b j e c t s i n e a c h$ $e t c .$ $(n = n_{1} m_{1} + n_{2} m_{2} + . . . + n_{r} m_{r})$ $i s \frac{n!}{\underset{k = 1}{\prod^{r}} {(m_{k}!)}^{n_{k}} n_{k}!}$ $t h e r e f o r e f o r e x a m p l e$ $3 / 2 / 1 \Rightarrow \frac{6!}{3! 2! 1!} = 60 = C (6, 3) C (3, 2) C (1, 1)$ $2 / 2 / 2 \Rightarrow \frac{6!}{{(2!)}^{3} 3!} = 15 \neq C (6, 2) C (4, 2) C (2, 2)$
	Answered by mr W last updated on 16/Oct/24

	$i f t h e b o x e s m a y n o t b e e m p t y :$ $S (6, 3) = 90 ✓$ $i f t h e b o x e s m a y b e e m p t y :$ $S (6, 3) + S (6, 2) + S (6, 1) = 90 + 31 + 1 = 122 ✓$
	Answered by mr W last updated on 17/Oct/24

	$6 / 0 / 0 \Rightarrow 1 w a y$ $5 / 1 / 0 \Rightarrow C_{1}^{6} = 6 w a y s$ $4 / 2 / 0 \Rightarrow C_{2}^{6} = 15 w a y s$ $3 / 3 / 0 \Rightarrow C_{3}^{6} / 2 = 10 w a y s$ $4 / 1 / 1 \Rightarrow \frac{6!}{4! {(1!)}^{2} 2!} = 15 w a y s$ $3 / 2 / 1 \Rightarrow \frac{6!}{3! 2! 1!} = 60 w a y s$ $2 / 2 / 2 \Rightarrow \frac{6!}{{(2!)}^{3} 3!} = 15 w a y s$ $t o t a l l y : 122 w a y s ✓$
Terms of Service
Privacy Policy
Contact: info@tinkutara.com

Question and Answers Forum