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Aim: What is the counting rule?

Aim: What is the counting rule?. Exam Tomorrow. Three Rules. Sometimes we need to know all possible outcomes for a sequence of events We use three rules 1. Fundamental counting rule 2. Permutation rule 3. Combination rule. Fundamental Counting Rule.

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Aim: What is the counting rule?

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  1. Aim: What is the counting rule? Exam Tomorrow

  2. Three Rules • Sometimes we need to know all possible outcomes for a sequence of events • We use three rules 1. Fundamental counting rule 2. Permutation rule 3. Combination rule

  3. Fundamental Counting Rule • In a sequence of n events in which the first one has k1 possibilities and the second event has k2 and the third has k3 and so forth, the total number of possibilities of the sequence will be k1 * k2 * k3 … kn • In each case and means to multiply

  4. Example • A coin is tossed and a die is rolled. Find the number of outcomes for the sequences of events. • Solution: Since the coin can land either heads up or tails up and since the die can land with any one of six numbers face up, there are 2 * 6 = 12 possibilities • A tree diagram can also be drawn for the sequences of events

  5. Repetition • When determining the number of possibilities of a sequence of events, one must know whether repetitions are permissible.

  6. Example • Example 1: The digits 0, 1, 2, 3, and 4 are to be used in a four-digit ID card. How many different cars are possible if repetitions are permitted? • Solution: 5 * 5 * 5 * 5 = 54 = 625 VS • Example 2: The digits 0, 1, 2, 3, and 4 are to be used in a four-digit ID card. How many different cars are possible if repetitions are not permitted? • Solution: 5 * 4 * 3 * 2 = 120

  7. Factorial Notation • Uses exclamation point • 5! = 5 *4 * 3 * 2 * 1 • For any counting n: n! = n(n-1)(n-2)(n-3)…1 • 0! = 1

  8. Permutations • Permutations: an arrangement of n objects in a specific order • The arrangement of n objects in a specific order using r objects at a time is called a permutation of n objects taking r objects at atime • It is written as nPr and the formula is

  9. Example • There are three choices: A, B, C of which need to fill in 5 spots. Repetition is allowed. • Solution:

  10. Combinations • Combinations: a selection of distinct objects without regard to order • The number of combinations of r objects selected from n objects is denoted by nCr and is given by the formula

  11. Example • How many combinations of 4 objects are there, taken 2 at a time? • Solution:

  12. Exam Tomorrow • Complete Review Sheet • Solutions will be on website • Study • In particular lessons 8 - 16

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