Math Is Fun Forum

tony123

Member

Registered: 2007-08-03

Posts: 230

In how many ways

In how many ways can one distribute 6 apples, 7 bananas, and
8 cantaloupes between 10 people? All fruit of the same type are identical.
Fruit must remain whole, must all be given out, and possibly Alphonse gets
everything.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: In how many ways

Distribute the apples A ways, then bananas B ways, then cantaloupes C ways (though I can't figure out why anyone would ever want a cantaloupe).  Then the answer will be A * B * C.

This is a balls in boxes problem, and this solution generalizes.  We shall represent each apple by the binary bit '1', and the boundary of a box by '0'.  So if there were only 3 people, one possible combination would be:

11001111

So that the first person gets 2 apples, the 2nd gets 0 and the 3rd gets 4.  Counting all such strings with 2 0's will give us the number of ways to distribute 6 apples.  Remember that the number of 0's is always the number of boxes (people) minus 1.  So this case, there are 10 people and 9 0's, and we are placing 6 1's.  Thus, the binary string has length 9 + 6 = 15.  Once we place the 9 0's (or equivalently, 6 1's), we have determined the binary string as all other positions must be 1's.  This is C(15, 9).  Notice that C(15, 9) = C(15, 6), so whether we place choose positions of 0's or 1's does not matter.

The final answer: C(15, 9) * C(16, 9) * C(17, 9)

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."