powerpoint presentation on illustrating combination of objects.pptx
- 1. Combinations Is the number of ways of choosing r items from a group of n items without regard to order or sequence. Order does not matter.
- 2. Permutation Combination Permutation refers to the different methods of arranging a set of objects in sequential order. Combination refers to the process of selecting items from a large set of objects, such that their order does not matter. It is used for things of different kinds. It is used for things of similar kinds. The arrangement of permutation is relevant. The arrangement of a combination is irrelevant. It denotes the arrangement of objects. It denotes the selection of elements instead of the arrangement of objects. It’s possible to derive multiple permutations from a single combination. From a single permutation, only a single combination can be derived. Permutation is referred to as an Ordered Set. Combination is referred to as Unordered Set. Permutation Formula is given as nPr=n!(n−r)!. Combination Formula is given as nCr=n!r!(n−r)!. It indicates different ways to arrange things, people, digits, alphabets, colors, etc. It indicates different ways of selecting menu items, food, clothes, subjects, etc. Example: Permutation of two letters from given letters a, b, c is ab, ba, bc, cb, ac, ca. Example: Combination of two letters from given letters a, b, c is ab, bc, ca.
- 3. Key Points About Combinations: 1.Order doesn't matter: In combinations, the order in which you select the items doesn't matter. For example, selecting Alice, Bob, Charlie is the same as selecting Charlie, Bob, Alice. 2.Without replacement: Once you choose an item, it is no longer available for selection again.
- 4. Remember: The combination of n objects taken r at a time is: ,
- 5. Basic Combinations Examples: 1. In how many ways can a committee consisting of 4 members be formed from 8 people? 2. A pizza can have 3 toppings out of a possible 7 toppings. How many different pizzas can be made? 3. How many polygons can be possibly formed from 6 distinct points on a plane, no three of which are collinear?
- 6. Combinations Including Specific Items Examples: 1. A school committee of 5 is to be formed from 12 students. How many committees can be formed if John must be on the committee? 2. From a deck of 52 cards, a 5 card hand is dealt. How many distinct five card hands are there if the queen of spades and the four of diamonds must be in the hand?
- 7. Combinations From Multiple Selection Pools Examples: 1. A committee of 3 boys and 5 girls is to be formed from a group of 10 boys and 11 girls. How many committees are possible? 2. From a deck of 52 cards, a 7 card hand is dealt. How many distinct hands are there if the hand must contain 2 spades and 3 diamonds?
- 8. At Least / At Most Examples: 1. A committee of 5 people is to be formed from a group of 4 men and 7 women. How many possible committees can be formed if at least 3 women are on the committee? 2. From a deck of 52 cards, a 5 card hand is dealt. How many distinct hands can be formed if there are at least 2 queens?
- 9. Permutations and Combinations Together Examples: 1. How many arrangements of the word TRIGONAL can be made if only two vowels and three consonants are used? 2. There are 7 men and 10 women on a committee selection pool. A committee consisting of President, Vice – President, and Treasurer is to be formed. How many ways can exactly two men be on the committee?
- 10. Example: 1. 12 people at a party shake hands once with everyone else in the room. How many handshakes took place? Other Types of Combinations Handshakes / Teams n C2 n = number of people or teams
- 11. Example: 1. A polygon has 7 sides. How many diagonals can be formed? Diagonals n C2 – n n = number of sides Multiple Combinations n Cr • k k = number of times all the possible combinations must happen 1. If each of the 8 teams in a league must play each other three times, how many games will be played?
- 12. Example: 1. If there are 15 dots on a circle, how many triangles can be formed? Making Shapes n Ck n = points in a circle, k = number of vertices Choosing One or More n = number of items in total 1. In how many ways ca you choose one or more of 12 different candies? 2n - 1