Section 4.5-Counting Rules

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Presentation transcript:

Section 4.5-Counting Rules Counting rules are useful in determining the number of possible outcomes for a given event Types of Counting Rules Fundamental Counting Rule Permutation Rule Combination Rule Section 4.5-Counting Rules

Fundamental Counting Rule For a sequence of events in which the first event can occur k1 ways and the second event can occur k2 ways, etc. the events together can occur a total of ways. page 156 of text

Notation The factorial symbol ! denotes the product of decreasing positive whole numbers. n! = n (n-1) (n-2)• • • (3) (2) (1) Special Definition: 0! = 1 Find the ! key on your calculator Use of a calculator with this topic will be necessary. page 157 of text

Factorial Rule A collection of n different items can be arranged or ordered in n! different ways.

(assuming all items are different) Permutations Rule (assuming all items are different) n is the number of available items r is the number of items to be selected Then, the number of permutations (or sequences) is P n! n r = Give some examples of permutation problems. Most calculators have this formula as a function. page 159 of text (n - r)! Note that order is important

Combinations Rule n! nCr = r!(n - r )! n different items r items to be selected The number of combinations is n! nCr = r!(n - r )! This formula will be helpful in Chapter 4 with the Binomial Probability Formula. Most calculators have this formula as a function. page 161 of text Note that order is NOT important

When different orderings of the same items are to be counted separately (as a separate outcome), we have a permutation problem, but when different orderings are not to be counted separately, we have a combination problem.

Suppose that you flip a coin a coin and roll a die Suppose that you flip a coin a coin and roll a die. Determine the number of possible outcomes to the experiment. Using the fundamental counting rule: Example

Suppose that you decide to choose 3 students from our class of 21 people to attend a statistics conference. How many possible sets of attendees are possible? Example

Suppose that you decide to award 2 customers a special gift card for visiting your store. One card is for $1000 and the other one is for $100. If you have 30 customers on a particular day, how many ways could you award the cards? Example