2005 PMWC Problems/Problem I6

Problem

A group of $100$ people consists of men, women and children (at least one of each). Exactly $200$ apples are distributed in such a way that each man gets $6$ apples, each woman gets $4$ apples and each child gets $1$ apple. In how many possible ways can this be done?

Solution

$m + w + c = 100$ $6m + 4w + c = 200$

Subtracting the second equation from the first, we get $5a + 3b = 100$ . Looking at this equation $\mod{5}$ , we see that $3b$ must be a multiple of 5, so $5|b$ . Thus the choices for $b$ are $5,10,15,20,25,30$ , which gives us $6$ possible choices.

2005 PMWC (Problems)
Preceded by Problem I5	Followed by Problem I7
I: 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 T: 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10

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2005 PMWC Problems/Problem I6

Problem

Solution

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