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We now come to our last major
class of counting problems.

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We will count the number of ways
that a given set can be

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partitioned into pieces
of given sizes.

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We start with a set
that consists

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of n different elements.

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And we have r persons.

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We want to give n1 items to
the first person, give n2

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items to the second
person, and so on.

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And finally, we want
to give n-sub-r

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items to the rth person.

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These numbers, n1, n2, up to nr
are given to us, how many

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items each person should get.

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And these numbers must add to
n so that every item in the

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original set is given
to some person.

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We want to count to the number
of ways that this can be done.

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This is the number of ways that
we can partition a given

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set into subsets of
prescribed sizes.

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Let's use c to denote
the number of

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ways this can be done.

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We want to calculate
this number c.

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Instead of calculating directly,
we're going to use

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the same trick that we employed
when we counted

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combinations and derived the
binomial coefficient.

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That is, we're going to
consider, in a much simpler

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counting problem, the problem
of ordering n items, taking

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the n items in our original
set and putting them in an

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ordered list.

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Of course, we know in how many
ways this can be done.

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Ordering n items can be done
in n factorial ways.

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This is the count
of the number of

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permutations of n items.

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But now let us think of a
different way of ordering the

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n items, an indirect way.

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It proceeds according to
the following stages.

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We start with the n items.

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And we first distribute them
to the different persons.

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Having done that, then we ask
person one to take their

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items, order them, and put
them in the first n1

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slots of our list.

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Then person two takes their
items and puts them into the

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next n2 slots in our list.

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We continue this way.

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And finally, the last person
takes the items that they

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possess and puts them
in the last n-sub-r

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slots in this list.

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In how many ways can this
process be carried out?

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We have c choices on how
to partition the

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given set into subsets.

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Then person one has n1 factorial
choices on how to

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order the n1 items that
that person processes.

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Person two has n2 factorial
choices for how to order the

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n2 items that it possesses,
and so on until the last

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person, who has nr factorial
choices for

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ordering their elements.

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This multi-stage process results
in an ordered list of

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the n terms.

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This is the number of ways these
multi-stage process can

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be carried out.

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On the other hand, we know that
the number of possible

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orderings of the items
is n factorial.

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So we have this equality.

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We can solve this for c.

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And we find the answer, that the
number of ways that the n

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items can be partitioned into
subsets of the given sizes is

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n factorial divided by the
product of the factorials of

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the different ni's.

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This particular expression
is called the multinomial

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coefficient, and it generalizes
the binomial

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coefficient.

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The binomial coefficient was
referring to the case where we

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essentially split our set into
one subset with k elements,

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and then the second subset gets
the remaining elements.

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So the special case where r is
equal to 2, and n1 is equal to

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k, n2 equals to n minus k,
this corresponds to a

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partition of a set into two
subsets, or what is the same

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just selecting the first subset
and putting everything

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else in the second subset.

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And you can check that in this
particular case, the

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expression for the multinomial
coefficient agrees with the

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expression that we had derived
for the binomial coefficient.