5 3 Permutations and Combinations Urn models We

5. 3 Permutations and Combinations • • Urn models We are given set of n objects in an urn (don’t ask why it’s called an “urn” - probably due to some statistician years ago). We are going to pick (select) r objects from the urn in sequence. After we choose an object -- we can replace it-(selection with replacement) -- or not -(selection without replacement). If we choose r objects, how many different possible sequences of r objects are there? Does the order of the objects matter or not? 1

Permutations • Selection without replacement of r objects from the urn with n objects. • A permutation is an arrangement. Order matters. • The number of permutations of n things taken r at a time P(n, r) = n(n - 1)(n - 2). . . (n - r + 1) • Note: P(n, r) = n! /(n - r)! • Example: Let A and B be finite sets and let | A | | B |. Count the number of injections from A to B. 2

Permutations • Theorem 1: If n is a positive integer and r is an integer with 1 r n, then there are P(n , r) = n(n-1)(n-2) ‥ (n-r+1) r- permutations of a set with n distinct elements. • Corollary 1: If n and r are integers with 0 r n , then P(n, r)= n! / (n-r)! 3

Permutations • A permutation of a set of distinct objects is an ordered arrangement of these objects. • We also are interested in ordered arrangements of some of the elements of a set. • An ordered arrangement of r elements of a set is called an r-permutation. • Example 2: let S={1, 2, 3}. The ordered arrangement 3, 1, 2 is a permutation of S. The ordered arrangement 3, 2 is a 2 -permutation of S. • Example 4: How many ways are there to select a firstprize winner, a second-prize winner, and a third-prize winner from 100 different people who have entered a contest? • HW: example 7, (p. 357) 4

Combinations • Selection is without replacement but order does not matter • It is equivalent to selecting subsets of size r from a set of size n. • Divide out the number of arrangements or permutations of r objects from the set of permutations of n objects taken r at a time: • The number of combinations of n things taken r at a time Other names for C(n, r): • n choose r • The binomial coefficient 5

Combinations • Theorem 2: The number of r-combinations of a set with n elements, where n is a nonnegative integer and r is an integer with 0 r n, equals C(n, r) = n! / r! (n-r)! • Corollary 2: Let n and r be nonnegative integers with r n. Then C(n, r) = C(n, n-r). 6

Combinations • Example: How many subsets of size r can be constructed from a set of n objects? • Corollary: r=0 n C(n, r)=2 n • Proof: • If we count the number of subsets of a set of size n, we get the cardinality of the power set. 7

Combinations • Example 11: How many poker hands of five cards can be dealt from a standard deck of 52 card? Also, how many ways are there to select 47 cards from a standard deck of 52 cards? • Example 13: A group of 30 people have been trained as astronauts to go on the first mission to Mars. How many ways are there to select a crew of six people to go on this mission ( assuming that all crew members have the same job)? • Example 14: How many bit strings of length n contain exactly r 1 s? • HW: Example 15, p(360) 8