What is the probability that in a class of 18 people, there exists exactly a group of exactly 3 people born on the same day of the week?


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Question Number 199353 by mr W last updated on 01/Nov/23

$W h a t i s t h e p r o b a b i l i t y t h a t i n a c l a s s$ $o f 18 p e o p l e, t h e r e e x i s t s e x a c t l y a$ $g r o u p o f e x a c t l y 3 p e o p l e b o r n o n t h e$ $s a m e d a y o f t h e w e e k ?$
Commented by nikif99 last updated on 03/Nov/23

$M y a p p r o a c h :$ $F r o m t h e o r y : d i s t r i b u t i o n s o f n$ $e l e m e n t s i n c c e l l s (w i t h 0 e l e m e n t s$ $a l l o w e d) = C_{n}^{n + c - 1} = C_{18}^{18 + 7 - 1} = 134596$ $F r o m p r o g r a m m i n g : c a s e s o f o n l y o n e$ $g r o u p o f o n l y 3 p e r s o n s = 51646$ $(c a s e s o f o n e o r m o r e g r o u p s o f o n l y$ $3 p e r s o n s = 77105) .$ $\Rightarrow p % = 51646 / 134596 = 38 . 37 %$
Commented by deleteduser1 last updated on 03/Nov/23

$T h e t o t a l n u m b e r o f d i s t r i b u t i o n o f 18 p e o p l e$ $i n t o 7 d a y s i s 7^{18} .$ $E a c h p e r s o n c a n b e b o r n i n o n e o f 7 d a y s$ $S o, 7 \times 7 \times 7 \times 7 . . . \times 7 (18 t i m e s) = 7^{18}$
Commented by deleteduser1 last updated on 03/Nov/23

$S o, i f y o u r p r o g r a m w e r e t o b e r i g h t, t h e c o r r e c t$ $a n s w e r m a y b e \frac{51646}{7^{18}}$
Commented by nikif99 last updated on 03/Nov/23

$Y o u a r e r i g h t . T h e f o r m u l a I u s e d$ $r e f e r s t o n o n d i s t i n c t i f e l e m e n t s .$ $F o r d i s t i n c t i f e l e m e n t s i s c^{n} .$
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