Permutations and Combinations

Fundamental Counting Principle If there are r ways of performing one operation, s ways of performing a second operation, t ways of performing a third operation, and so on… then the number of ways of performing the operations in succession are: Three different mathematics books and five other different books are to be arranged on a bookshelf. Find out the number of possible arrangements if the three mathematics books must be kept together. Answer: 4320 ways

A B C D E A B C D E is the same as: Circular Arrangements If we are looking for the number of arrangements of items around a circular table, we have to consider that some arrangements may be exactly alike. This arrangement: Therefore, to solve these circular arrangements, you fix one of the elements and arrange the remaining items around it. The number of arrangements of n unlike things in a circle will be:

If clockwise and anti-clockwise arrangements are considered identical, then the number of arrangements would be:

Four men are to be seated at a circular table. In how many ways can this be done? Answer: 6 ways Nine beads, all of different colors, are to be arranged on a circular wire. Two arrangements are not different if they are the same when the ring is turned over. How many different arrangements are possible? Answer: ways

The letters of the word TUESDAY are arranged in a line, each arrangement ending with the letter S and starting with the letter D. In how many ways is this possible? Answer: 120 ways How many numbers greater than can be formed using the digits 2, 3, 4, 5 and 6 if each digit is used only once in each number? Answer: 72 ways

Mutually Exclusive Situations Two situations are mutually exclusive if situation A occurs, then situation B cannot occur. Likewise, if situation B occurs, then situation A cannot occur. Therefore: or

Sheila, Ann and Harvey are fifth year students. Sarah, Jeff and John are fourth year students and Alan, Chris and Mark are third year students. In how many ways can these pupils be arranged in a line if the pupils from each year are kept together? Answer: 1296 In how many ways can six boys and two girls be arranged in a line if the two girls must not sit together? Answer: 30240

If we want to choose r items from a group of n objects, we would first have n choices, then (n – 1) choices, then (n – 2) choices and so on, until the r th choice. This can be solved as: This is normally called a permutation and is usually written as:

Arrangements of Like and Unlike Things Suppose the following letters are arranged in a row: We already know that there would be 5! possible arrangements. Now suppose we cannot distinguish between each like letter. So, we want to arrange the letters: There will bepossible arrangements.

Therefore, the number of arrangements of n things with p of one kind, q of another kind, r of another and so on, will be: In how many ways can 4 red, 3 yellow and 2 green disks be arranged in a row, if disks of the same color are indistinguishable? Answer: 1260 ways

Find the number of three-letter arrangements that can be made from the letters of the word PYTHAGORAS. How many different three-digit even numbers can be formed from the figures 2, 5, 7 and 9 if repetitions are not allowed. Answer: 528 Answer: 6

In how many ways can the letters of the word PARALLEL be arranged in a row? In how many ways can the letters of the word BANANA be arranged with the N’s separated? Answer: 3360 Answer: 40

Find the number of two-letter arrangements made from the letters A, B, C and D. We can see that there are 12 arrangements: AB, AC, AD, BA, BC, BD, CA, CB, CD, DA, DB, DC But, notice that many of these arrangements are the same: AB = BA, AC = CA, AD = DA, BC = CB, BD = DB, DC = CD So, the number of combinations of two-letters is one-half of 12, or 6

We know that an arrangement, or permutation, of n things taken r at a time is defined as: But, each selection of r items will have r! items that are the same. We must remove these r! identical items. Therefore, the number of combinations of n objects taken r at a time is: Use a permutation when the order of an arrangement is important. Use a combination when the order is NOT important.

Find the number of selections of 4 letters that can be made from the letters of the word SPHERICAL. How many of these selections do not contain a vowel? Answer: 126 Answer: 15 How many different committees, each consisting of 3 boys and 2 girls, can be chosen from 7 boys and 5 girls? Answer: 350

A team of 7 players is to be chosen from a group of 12 players. One of the 7 is then to be elected as captain and another as vice-captain. In how many ways can this be done? Answer: A group consists of 4 boys and 7 girls. In how many ways can a team of five be selected if it is to contain at least one member of each sex. In how many ways can a team of five be selected from 4 boys and 7 girls if the team is to contain at least 3 boys. Answer: 441 Answer: 91

To find the number of selections of any size from a group, you have to consider every possibility. Example: How many different selections can be made from the five letters a, b, c, d, e? First Method: Total is 31 Second Method: There are two choices for the letter a. Either it is included or not. Likewise, there are two choices for b and also for c, d, and e. So, there are: choices. But, we must remove the case where no letters are included. So, there are:

Divisions into Groups Example: The letters a, b, c, d, e, f, g, h and I are to be divided into three groups containing 2, 3 and 4 letters respectively. In how many ways can this be done? Answer: In general, the number of ways of dividing (p + q + r) unlike things into three groups containing p, q and r things respectively is:

Find the number of ways that 12 people can be arranged into groups if there are to be: a) Two groups of 6 people. b) 3 groups of four people. Answer: 462 ways Answer: 5775 ways

How many different whole numbers are factors of the number below? Answer: 64 How many different selections can be made from the letters of the word INABILITY? Answer: 255

How many subsets of a set of 10 elements have either 3 or 4 elements? Answer: 330 To participate in a state lottery, a person must choose 4 numbers from 1 to 40. In how many ways can this be done if: a) the order of the choice matters b) the order of the choice does NOT matter. Answer: a) b) 91390

There are 10 seats in a row in a waiting room. There are six people in the room. (a) In how many different ways can they be seated? (b) In the group of six people, there are three sisters who must sit next to each other. In how many different ways can the group be seated? Answer: (a) (b) M06/HL1/19