Permutations and Combinations Essential Ideas for Sections 13.1 Permutations and Combinations

Fundamental Counting Principle If a task consists of a sequence of choices in which there are p selections for the first choice, q selections for the second choice, r selections for the third choice, and so on, then the task of making these selections can be done in different ways.

Fundamental Counting Problem Suppose you go to a sub shop to buy a sandwich. There are 5 varieties of bread, 6 varieties of meat, and 3 varieties of cheese. How many different types of sandwiches are possible? 5·6·3 = 90

Factorial Notation For any counting number n, the factorial of n is defined by: Also, we define 0! = 1

Factorial Computations

Countdown Property The countdown property allows us to simplify fractional expressions involving factorials. Countdown Property: Example:

Permutations A permutation is an ordered arrangement of n distinct objects without repetitions. The symbol P(n, r) represents the number of permutations of n distinct objects, taken r at a time, where r < n.

Number of Permutations of n Distinct Objects Taken r at a Time The number of different arrangements from selecting r objects from a set of n objects (r < n), in which 1. the n objects are distinct 2. once an object is used, it cannot be repeated 3. order is important (Election problem) is given by the formula

Permutation Example

In how many ways can a class of twenty students elect a president, vice president, and secretary? This problem can be solved using the fundamental counting theorem or using the permutation formula: As a counting problem: As a permutation:

Distinguishable Permutations The number of distinguishable permutations of n objects, in which one or more of the objects are indistinguishable, can be found by dividing n! by the product of the total number of permutations for each indistinguishable object.

Find the number of permutations of the letters in the word ATTRACT There are seven objects, but the repeated A’s and T’s are indistinguishable. Since there are two A’s and three T’s, the solution is found as follows:

Combinations A combination is an arrangement, without regard to order, of n distinct objects without repetitions. The symbol C(n, r) represents the number of combinations of n distinct objects taken r at a time, where r < n.

Number of Combinations of n Distinct Objects Taken r at a Time The number of different arrangements from selecting r objects from a set of n objects (r < n), in which 1. The n objects are distinct 2. Once an object is used, it cannot be repeated 3. Order is NOT important (Committee problem) is given by the formula:

Combination Example

In how many ways can a class of twenty students elect a committee of three? Using the combination formula:

In how many ways can a full house of three tens and two queens be dealt? There are four tens to choose from and we want to draw three tens; also, there are four queens to choose from and we want to draw two queens so using the combination formula along with the fundamental counting principle gives us:

In how many ways can a club with 20 members elect a president, vice president, and secretary, and then from the remaining members select a committee of five people? The first part of this problem is a permutation because order is important and the second part is a combination because order is not important. Using the permutation and combination formulas along with the fundamental counting principle gives us: