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296
A Combinatorial Triad
Cyrus Hsia
student, University ofToronto
One of the beautiful aspects of mathematics is that solving a problem
can be done in so many ways. In combinatorics, the study of counting and
ways of counting, this is more than evident in its many interesting problems.
See any standard textbook on combinatorics [1] to convince yourself. Consider
the following three standard combinatorial problems and see if you can
nd the link between them.
Problem 1. What is the number of ways to distribute 10 identical apples
into 3 baskets labelled A, B, and C?
Problem 2. How many ways are there to select 10 ice-cream scoops
from 3 di erent avours onanice-cream cone?
Problem 3. Find the number of non-negative integer solutions to the
diophantine equation x1 + x2 + x3 =10.
If you said the answersare all the same, then you arecorrect. The three
problems above are di erent ways of counting something equivalent. Each
way of thinking may be more helpful in certain situations than others. The
following are solutions to approaching the apparently di erent, but similar,
problems.
Solution to Problem 1. The standard approach to solving this type of
problem rests on the following key trick. Let the apples be represented by
1's since they are all identical. Now use two 0's to place between the 1's
to split them into 3 piles. Put the rst pile into basket A, the second into
basket B and the remaining pile into basket C. The problem then reduces
to nding the number of ways of placing the 0's among the 1's.
There are10+1 = 11positions to place the rst 0, namely 9 spots
between the 10 1's and the front and back of the list of 1's. Now there are
11 numbers and 11+1 = 12positions to place the second 0. However,
we are counting everything twice! If we place 0 in front and then a 0 after
the fth 1, this is the same as if we place the rst 0 after the fth 1 and the
second 0 in the rst spot. Hence, there are
ways in total.
11 12
2
Solution to Problem 2. This is exactly the same as Problem 1 if we
consider the ice-cream scoops to be1's and the type of avour it is by its
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