{Combinatorics} Coins distributed among people.

[youngstudent16]

Homework Statement

Find the number of ways to distribute 55 identical coins among three people, so that everyone gets an odd number of coins.

Homework Equations

Stars and Bars Formula [/B]

The Attempt at a Solution

(n+r-1,n-1)

Ways to place r indistinguishable objects into n distinguishable boxes.
C(57,2)=1596 total ways [/B]


Thats about it. If it was even I could use same formula I think but with groups of objects instead. Since its odd I'm unsure.

My weak guess was I took 55/3 and used the same formula and got roughly 196 total ways I'm sure that is wrong though. Thanks for any help.

 

[youngstudent16]

My new attempt which came out right

My attempt thinking of it as solutions x1+x2+x3=55 solutions to the each such that each is odd z1+z2+z3=28 with no restrictions is the same as y1+y2+y3=52 such that each solution is even

Thus x1=y1+1=2z1+1

So number of ways will be 378

 

[geoffrey159]

I agree with your answer but don't follow your reasoning !
Mine is that if you've got $p_1,p_2,p_3$ coins in each box, all these numbers being odd, you can write $p_i = 2q_i - 1, \ q_i \ge 1$. Therefore your problem is equivalent to $q_1 + q_2 + q_3 = 29$ which is a classic situation of 'stars and bars' theorem.

 

[haruspex]

z1+z2+z3=28

You mean =26, right? The 2 gets added later.