How-many-ways-can-50-people-be-divided-into-3-groups-so-that-each-group-contains-members-equal-to-a-prime-number-

Question Number 160980 by cortano last updated on 10/Dec/21

How many ways can 50 people

be divided into 3 groups, so

that each group contains members

equal to a prime number ?

Answered by Rasheed.Sindhi last updated on 10/Dec/21

Sum of three primes = 50

\Rightarrow One is even prime and the remaining

two are odd primes

So one group is of 2 - people

The remaining 48 should be divided

into two^{'} prime groups .

48 = 7 + 41, 11 + 37, 17 + 31, 19 + 29

H e n c e 4 w a y s .

(2, 7, 41), (2, 11, 37), (2, 17, 31), (2, 19, 29)

Commented by cortano last updated on 10/Dec/21

not 30 ways ?

Commented by Rasheed.Sindhi last updated on 10/Dec/21

I t h i n k a l l t h e p e r m u t a t i o n s o f (2, 7, 41)

o n e w a y .

Commented by bobhans last updated on 10/Dec/21

how for (2, 5, 43) ?

Commented by Rasheed.Sindhi last updated on 10/Dec/21

Y e s t h i s a l s o . T h a n k y o u!

Answered by mr W last updated on 10/Dec/21

Commented by mr W last updated on 10/Dec/21

a + b + c = 50 w i t h a, b, c \in P

e x c e p t 2 a l l p r i m e n u m b e r s a r e o d d .

t h e s u m o f t h r e e o d d n u m b e r s i s

a l w a y s o d d . s u c h t h a t t h e

s u m o f t h r e e p r i m e n u m b e r s i s 50,

o n e a n d o n l y o n e o f t h e m m u s t b e 2 .

s a y a = 2, t h e n b + c = 48 . f r o m t h e

p r i m e n u m b e r t a b l e a b o v e w e s e e

t h e r e a r e f o l l o w i n g p o s s i b i l i t i e s :

43 + 5

41 + 7

37 + 11

31 + 17

29 + 19

t o d i v i d e 50 p e o p l e i n t h r e e g r o u p s

w i t h a, b, c p e o p l e i n e a c h g r o u p

r e s p e c t i v e l y t h e r e a r e \frac{50!}{a! b! c!} w a y s .

s o t h e t o t a l n u m b e r o f w a y s i s :

\frac{50!}{2! 5! 43!} + \frac{50!}{2! 7! 41!} + \frac{50!}{2! 11! 37!} + \frac{50!}{2! 17! 31!} + \frac{50!}{2! 19! 29!}

\approx 1.94 \times 10^{16}

Commented by Tawa11 last updated on 10/Dec/21

Great sir

Commented by cortano last updated on 10/Dec/21

thank you

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