PERMUTATIONS AND COMBINATIONS

PERMUTATIONSANDCOMBINATIONSWhat is permutation?A permutation of a set of elements is an arrangement of a specified number of those elements in a definite order.Remember: Order matters a lot when we are arranging the elements of a given set.List all the possible two-letter permutations of A, B and CAB, AC, BC, BA, CA, CB

How many have we listed? six

Note: If were given not just three letter, say 10, it will take us long time to list all permutations.

Kinds of permutations:1) Linear permutationA permutation of different elements in a line or in row.

2) Distinguishable permutationA permutation of elements where some are identical or alike.

3) Circular permutationA permutation of all elements in a set that follows a cyclic pattern

Linear PermutationsExamples: 3. In how many ways can four persons pose for a picture if they are to stand in a row?

The order by which the four persons pose matters and each of them will be in the picture, so the answer is P(4, 4) = 4 x 3 x 2 x 1 = 24 ways4. How many three-digit numbers can be formed using the digits 3, 4, 5, 6, 7, and 8 if there is no repetition of the digits?P(6, 3) = 120 numbers

5. How many ways are there for nine students to fall in line if two particular students refuse to be parted?a) Think of the two particular students as a single person. P(8, 8) = 40 320b) Allow the two students to switch places and use the rule of product. P(8, 8) x 2 = 80 640 ways

Distinguishable PermutationExamples:Circular permutationWhat is Combination?A combination is the number of arrangements of a set of observations such that the order is not considered.

Determine which is a permutation or combination in the following cases:1. Choosing a team of 4 from a group of 12.2. Choosing a chairman, co-chairman, secretary and a treasurer from a group of 12. 3. From 15 sophomores, how many social action groups can be formed consisting of 4 sophomores?C(15,4) = 1, 365 groups

4. From 6 boys and 4 girls, how many committees can be formed consisting of 2 boys and 3 girls?C(6,2) x C(4,2) = 15 x 4 = 60 committees Exercises:Determine if the situation involves combination or permutation.Picking the names of three students from a class to attend a meetingSelecting the president and vice-president of the classChoosing four members of a committee from a group of 15 personsGetting seven tiles in a scrabble gameObtaining tickets to win first, second, and third prizes in a raffle draw6. Drawing four cards from a deck of 52 cards7. Assigning students to their seats in a mathematics class8. Selecting three teachers to attend a seminar9. Arranging three different vases on a tray10. Inviting eleven of your close friends to a partyB. Compute the following:

P(7, 2) (b) P(8, 8) (c) P(25, 2) A baseball team has nine players. Find the number of ways the manager can arrange the batting order. There are five members of the Math Club. In how many ways can the positions of officers, a president and a treasurer, be chosen? a) C(7,2) (b) C(8,8) (c) C(25,6) The Library of Science Book Club offers three books from a list of 42. If you circle three choices from a list of 42 numbers on a postcard, how many possible choices are there? There are five members of the math club. In how many ways can the two person Social Committee be chosen?