Permutations & Combinations Section 2.4

Permutations When more than one item is selected (without replacement) from a single category and the order of selection is important, the various possible outcomes are called permutations. The notation used for permutations is n P r. The number of permutations of r items selected without replacement from a pool of n items (note r ≤ n) is given by: n P r =

Example 1: Choosing a Committee of 4 members where order matters The Board of Directors of XYZ Corporation has 15 members. In how many ways can we choose a committee of 4 members where we have a President, Vice-President, Treasurer and Secretary? Here we have a situation where order matters. The pool of candidates is 15, that is n = 15. The number of seats on the committee is 4, hence r = 4. The number of committees would be:

Combinations When one or more item is selected (without replacement) from a single category and the order of selection is not important, the various outcomes are called combinations. The notation used for combinations is. The number of combinations of r items selected without replacement from a pool of n items (note r ≤ n) is given by:

Example 2: Choosing a Committee of 4 members where order does not matter The Board of Directors of XYZ Corporation has 15 members. In how many ways can we choose a committee of 4 members, (each member has equal rank)? Here order does not matter. The pool of candidates is 15, that is n = 15. The number of seats on the committee is 4, hence r = 4. The number of committees would be:

Permutations of identical items Say you want all the distinct permutations of the word SEE. How many do you have? Colorize the two Es. Then list the permuations SEE, SEE, ESE, ESE, EES, EES If you take away the color you get repeats: SEE, SEE, ESE, ESE, EES, EES The only distinct permutations are SEE, ESE, EES, i.e. 3. The formula for the number of distinct permuations is Where n is the total number of items and x is the number of times the first item is repeated, y is the number of times the second item is repeated, z is the number of times the third item is repeated etc. Example: SEE, n = 3, (Number of S) x = 1, and (Number of E) y = 2. Example: How many distinct permutations are there for the word MISSISSIPPI? ANSWER: n = 11, w = 1(M), x = 4 (I), y = 4 (S), z = 2 (P):