Counting Methods Topic 7: Strategies for Solving Permutations and Combinations

I can solve a contextual problem that involves permutations. I can solve a contextual problem that involves combinations.

Information Before beginning to solve a counting problem, it must be determined if order is important or not. If order is important, you can use the permutation formula or FCP. If order is not important, you can use the combination formula. Remember: A permutation is an arrangement in which order matters. Order may be implied by: physical position (ie. the number of ways 6 people can sit in 6 seats in a movie theatre) different title (ie. from 6 candidates the number of ways the position of president, secretary and treasurer can be filled) sequence (ie. the number of ways 8 horses can finish in 1 st place, 2 nd place and 3 rd place) words and numbers read left to right (ie. arranging the letters in the word VICTORIA)

Example 1 Classify each situation as either a permutation or a combination. Provide your reasoning for each situation. a) The number of ways of awarding a gold, silver and bronze medal to 6 competitors in an Olympic weightlifting competition. b) The number of 5 card-poker hands from a standard deck of 52 cards. c) The number of ways of picking two co-captains from a soccer team with 18 players. Classifying a permutations vs. a combination permutation – order matters, since each person is awarded a different medal combination – order of the cards does not matter combination – the order these two co-captains are selected makes no difference to the composition

Example 1 d) The number of 5-digit natural numbers that can be made using the digits 2, 3, 5, 7 and 9. e) The number of ways 3 people can sit in an ETS bus with 5 vacant seats. f) The number of ways of picking 3 winning entries of $1 000 each from 80 entries. permutation – order matters, since each arrangements of digits has a different value permutation – order matters since each person has to sit in a particular place combination – the order doesn’t matter since each person selected gets the same prize

Example 1 g) The number of ways a football team can run a 1 st down play and then a 2 nd down play from 7 possible offensive plays. h) The number of ways of selecting a pizza with 3 toppings from a menu that lists 12 possible toppings. permutation – order matters since the plays are the 1 st down play and the 2 nd down play combination – order doesn’t matter since a ham and pineapple pizza is no different than a pineapple and ham pizza

Example 2 A Tae Kwon Do instructor and her students are having a group photograph taken. There are three boys and five girls. a) The photographer wants the boys to sit together and the girls to sit together for one of the poses. How many ways can the students and instructor sit in a row of nine chairs for this pose? Solving a counting problem with conditions Since we are arranging the students in positions, order matters. Use permutations or FCP. The 3 boys are a single group with 3! arrangements. Now there are 3 objects to arrange: 1 group of girls, 1 group of boys and 1 instructor Within the group of boys there are 3! arrangements and within the group of girls there are 5! arrangements. There are 3 groups to arrange. The 5 girls are a single group with 5! arrangements. One instructor.

Example 2 b) For another pose, the photographer wants the two tallest students, Jarod and Jodi, to sit at either end, Jarrod on the left and Jodi on the right, and the teacher to sit in the middle. How many different seating arrangements are there for this pose? Solving a counting problem with conditions ___ ___ ___ ___ ___ ___ ___ ___ ___ = Jarrod 1 Jodi 1 Only the teacher can sit here. 1 There are students left to fill the 6 remaining spaces (6!) ! = 720

Example 3 Solving a counting problem involving cases A standard deck of playing cards consists of 52 cards: 26 black and 26 red. How many different five-card hands that contain at most one black card can be dealt to one person from a standard deck of playing cards? Since we are selecting cards for a hand from a deck of cards, order doesn’t matter. Use combinations We want 5-card hands with at most 1 black card. This means either 0 black cards or 1 black card. 0 black cards (and 5 red) 1 black cards (and 4 red) OR

Example 4 Solving a Counting Problem Using the Fundamental Counting Principle From 8 students on Student Council, in how many ways can you select a President, a Treasurer and 3 students to be on the Social Committee? President AND Treasurer AND Others One student selected for President so only 7 students left to select from for Treasurer. Two students selected for President and Treasurer so only 6 students left to select from for the other 3 members.

Need to Know When solving counting problems, you need to determine if order plays a role or not to determine if you will be using permutations or combinations.

Need to Know Once you have determined if a problem involves permutations or combinations, Look for conditions and address these first. If there is a repetition of a of the n objects to be eliminated, it is usually done by dividing by a! If a problem involves multiple tasks that are connected by the word AND, then the fundamental counting principle can be applied. Multiply the number of ways each task can occur. If a problem involves multiple tasks that are connected by the word OR, the fundamental counting principle does not apply. Instead, add the number of ways that each task can occur. This typically is found in counting problems that involve several cases. You’re ready! Try the homework from this section.