Section 4.4 Binomial Identities
Each of the following problems provides a specific problem modeling an identity involving binomial coefficients.Problem 4.4.1.
Acme corporation is selecting members for a user feedback panel. The panel will consist of
How many ways can
be selected fromHow many ways can
be selected if Wile E. Coyote is one of them?How many ways can
be selected if Wile E. Coyote is not one of them?State and explain an identity relating part 1 to parts 2 and 3. Note this is one proof of the identity
Prove this identity using algebra.
If Wile E. Coyote is not in the set, then (n-1) choose k.
For all possible choices, add both together, to get the universal set.
Problem 4.4.2.
Allen and Betty have different management styles. Allen selects
How many ways can Allen select his way?
How many ways can Betty select her way?
State and explain a relation between the previous. Note this is one proof of the identity
Prove the relation using algebra.
1) n choose k times k choose m, or
From n people total, choose k employees. Then for each way this can be achieved, choose m leaders out of k employees.
2) n choose m times
This can be rewritten:
From n people total, choose m leaders. Then for each way this can be achieved, select from the remaining
3) The ways Allen and Betty select employees have the same result. The only difference is the order β Allen first chooses k employees, then chooses m leaders from the k, and Betty first chooses m leaders, then adds more employees to make k total. In both cases, n is the total number of people, there are m leaders and k employees, and we did not adjust any of these numbers for either method.
We will prove that the two methods are equal by beginning with the second method and showing that it is equal to the first method using algebra.
Problem 4.4.3.
βBlock Walkingβ is a model for binomial coefficients. Use the provided block walking diagram for the following problems. Coordinates below are
Write down all the ways to get from
to by recording whether one walks left or right from each intersection.Explain how the number of ways to get from
to could be counted without listing all of the routes.
Problem 4.4.4.
Use the block walking model for the following problems.
Count the number of right moves necessary to get from
to that start by going left first.Count the number of additional right moves necessary to get from
to that start by going right first.Explain how these two counts relate to the total number of ways to get from
toUse this to prove the identity
Problem 4.4.5.
Prove the following identities using the block walking method.