Discrete Mathematics- An Open Introduction - 3rd Edition, 2016a

118 1. Counting
counting the number of ways to assign elements from one set to elements
of another.
Example 1.6.6
You decide to give away your video game collection so as to better
spend your time studying advanced mathematics. How many ways
can you do this, provided:
1. You want to distribute your 3 different PS4 games among 5
friends, so that no friend gets more than one game?
2. You want to distribute your 8 different 3DS games among 5
friends?
3. You want to distribute your 8 different SNES games among 5
friends, so that each friend gets at least one game?
In each case, model the counting question as a function counting
question.
Solution.
1. We must use the three games (call them 1, 2, 3) as the domain
and the 5 friends (a,b,c,d,e) as the codomain (otherwise the
function would not be defined for the whole domain when
a friend didn’t get any game). So how many functions are
there with domain {1, 2, 3} and codomain {a, b, c, d, e}? The
answer to this is 5 3 125, since we can assign any of 5 elements
to be the image of 1, any of 5 elements to be the image of 2
and any of 5 elements to be the image of 3.
But this is not the correct answer to our counting problem,
( ) 1 2 3
because one of these functions is f ; one friend
a a a
can get more than one game. What we really need to do is
count injective functions. This gives P(5, 3) 60 functions,
which is the answer to our counting question.
2. Again, we need to use the 8 games as the domain and the 5
friends as the codomain. We are counting all functions, so the
number of ways to distribute the games is 5 8 .
3. This question is harder. Use the games as the domain and
friends as the codomain (the reverse would not give a function).
To ensure that every friend gets at least one game means that
every element of the codomain is in the range. In other words,
we are looking for surjective functions. How do you count
those??