Math 141 Lecture Notes
Section 6.4 Permutations and Combinations
Permutations
A permutation of a set of distinct objects is an arrangement of these objects in a definite order.
Example 1: Let A = {a, b, c}.
a. Find the number of permutations of A.
b. Use a tree diagram to list all the permutations of A.
Example 2: Find the number of ways a baseball team consisting of nine people can arrange themselves in a line for a group picture.
For any natural number n, n-factorial is given by
n! = n(n – 1)(n – 2)× . . . × 3 × 2 × 1
0! = 1
The number of permutations of n distinct objects taken n at a time:
P(n, n) = n!
The number of permutations of n distinct objects taken r at a time:
P(n, r) =
Example 3: Compute P(4, 4) and P(4, 2), and interpret your results.
Example 4: Let A = {a, b, c, d}.
a. Compute the number of permutations of the set A taken two at a time.
b. Use a tree diagram to display the permutations of part a.
Example 5: Find the number of ways a chairman, a vice-chairman, a secretary, and a treasurer can be chosen from a committee of eight members.
Permutations of n Objects, Not All Distinct:
Given a set of n objects in which objects are alike and of one kind, objects are alike and of another kind,…, and, finally, objects are alike and of yet another kind so that
then the number of permutations of these n objects taken n at a time is given by
Example 6: Find the number of permutations that can be formed from all the letters in the word ATLANTA.
Example 7: Weaver and Kline, a stock brokerage firm, has received nine inquiries regarding new accounts. In how many ways can these inquiries be directed to three of the firm’s account executives if each account executive is to handle three inquiries?
Combinations
A combination is a subset of r objects taken from a set of n objects, without any regard to the order in which the objects are selected.
The number of combinations of n distinct objects taken r at a time is given by
C(n, r) = (where r £ n)
Example 8: Compute and interpret the results of C(4, 4) and C(4, 2).
Example 9: A Senate investigations subcommittee of four members is to be selected from a Senate committee of ten members. Determine the number of ways this can be done.
Example 10: How many poker hands of 5 cards can be dealt from a standard deck of 52 cards?
Example 11: The members of a string quartet composed of two violinists, a violist, and a cellist are to be selected from a group of six violinists, three violists, and two cellists, respectively.
a. In how many ways can the string quartet be formed?
b. In how many ways can the string quartet be formed if one of the violinists is to be designated as the first violinists and the other is to be designated as the second violinist?
Example 12: Suppose an investor has decided to purchase shares in the stocks of two aerospace companies, two energy development companies, and two electronics companies. In how many ways may the investor select the group of six companies for the investment from the recommended list of five aerospace companies, three energy development companies, and four electronics companies?
Example 13: The Futurists, a rock group, are planning a concert tour with performances to be given in five cities: San Francisco, Los Angeles, San Diego, Denver, and Las Vegas. In how many ways can they arrange their itinerary if
a. There are no restrictions?
b. The tree performances in California must be given consecutively?
Example 14: The U.N. Security Council consists of 5 permanent members and 10 nonpermanent members. Decisions made by the council require nine votes for passage. However, any permanent member may veto a measure and thus block its passage. In how many ways can a measure be passed if all 15 members of the Council vote (no abstentions)?