We explore the Multinomial Theorem.
Consider the trinomial expansion of . The terms will have the form where , such as and . What are their coefficients? The coefficient of the first of these is the number of permutations of the word , which is and the coefficient of the second is These are multinomial coefficients and they are denoted respectively. Note that in this notation, ordinary binomial coefficients could be written as
The general multinomial coefficient is defined as where are non-negative integers satisfying
example 1 How many ways can a group of 10 people be broken into three subgroups
consisting of 2, 3 and 5 people?
First, we can select the subgroup of 2 people in ways. Then we can select the subgroup of 3 people in ways. finally, we can select the subgroup of 5 people in ways. By the FPC, the total number of ways to create the three subgroups is We can see this directly by lining up the 10 people and assigning each of them a 2, 3 or a 5 to determine their subgroup. The total number of such assignments is the number of permutations of the digits which is the indicated multinomial coefficient.
First, we can select the subgroup of 2 people in ways. Then we can select the subgroup of 3 people in ways. finally, we can select the subgroup of 5 people in ways. By the FPC, the total number of ways to create the three subgroups is We can see this directly by lining up the 10 people and assigning each of them a 2, 3 or a 5 to determine their subgroup. The total number of such assignments is the number of permutations of the digits which is the indicated multinomial coefficient.
Multinomial Theorem For a natural number and real numbers we have where
the sum runs over all possible non-negative integer values of whose sum is
.
example 2 Find the coefficient of in the expansion of .
According to the Multinomial Theorem, the desired coefficient is
According to the Multinomial Theorem, the desired coefficient is
(problem 2) Find the coefficient of the given term of the multinomial expansion:
a) in
b) in
c) in
d) in
a) in
b) in
c) in
d) in
Multinomial Identities
example 3 Find the sum of all multinomial coefficients of the form .
If we let and in the expansion of , the Multinomial Theorem gives where the sum runs over all possible non-negative integer values of and whose sum is 6. Thus, the sum of all multinomial coefficients of the form is .
If we let and in the expansion of , the Multinomial Theorem gives where the sum runs over all possible non-negative integer values of and whose sum is 6. Thus, the sum of all multinomial coefficients of the form is .
(problem 3) Find the indicated sum.
a) The sum of all multinomial coefficients of the form .
b) The sum of all multinomial coefficients of the form .
a) The sum of all multinomial coefficients of the form .
b) The sum of all multinomial coefficients of the form .
Sum of Multinomial Coefficients In general, where the sum runs over all
non-negative values of whose sum is .
- Proof
- The result follows from letting in the multinomial expansion of .
- Proof
- The result follows from a double counting argument for the number of ways to select subgroups of size from a group of size where . The left hand side of the identity gives this directly. The right hand side is obtained by considering one of the people as special and partitioning the collection of all subgroups according to which subgroup the special person is a member of.