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PreCalc Notes 9.6 Counting Principles Fundamental Counting Principle Explore. Draw a picture to complete the problem. Jordan was planning a birthday party for her best friend. She had to pick a set menu of one appetizer, one main dish, and one dessert. She had 4 appetizers, 5 main dishes, and two desserts to choose from. How many different options for her set menu does she have? The Fundamental Counting Principle โ AKA The Multiplication Counting Principle The number of possible outcomes of an event is equal to the product of the number of options at each stage of the event. Basicallyโฆ.. Example 1. Use the fundamental counting principle to solve each problem. A) How many different pairs of letters from the English alphabet are possible? B) Telephone numbers in the US have 10 digits. The first three digits are the area code, and the next seven digits are the local telephone number. How many different telephone numbers are possible within each area code? (Note: a local telephone number cannot begin or 0 or 1)
C) A combination lock will open when the right choice of five numbers (from 1-20, inclusive) is selected. How many difference lock combinations are possible? D) In a certain state, each automobile license plate number consists of two letters followed by a four- digit number. How many distinct plate numbers can be formed? E) How many different distinct plate numbers from example E could be formed if letters and numbers cannot be repeated? Permutations Definition of Permutation A permutation of ๐ different elements is an ordering of the elements such that one element is first, one is second, one is third, and so onโฆ. Basicallyโฆโฆ. Number of Permutations of ๐ Elements Example 2. Use your knowledge of the fundamental counting principle to help you solve each problem. A) How many permutations of the letters A, B, C, D, E, and F are possible? B) How many ways can eight children line up in a row? Number of Permutations of ๐ elements The number of permutations of n elements is: ๐ โ (๐ โ 1 ) โโโโ 4 โ 3 โ 2 โ 1 = ๐! In other words, ๐ elements can be ordered in ______ ways
Example 3. Find how many distinguishable permutations are possible for each situation. A) How many ways can the letters in the word BANANA be written? B) How many ways can the letters in the word TERRITORIAL be written? Combinations When we are counting combinations, we are counting groups of objects for which order does not matter in the situation. For instance, when choosing a team, the order in which the members does not make a difference in the composition of the team. Combinations of ๐ elements taken ๐ at a Time
nCr =^
๐! (๐โ๐)!๐! Example 4. Find the number of combinations possible in each situation. A) In how many different ways can three letters be chosen from the letters A, B, C, D, and E? B) In how many different ways can a 5 member group be chosen in a class of 22 students? C) How many different 4-topping pizzas can be made with the choice of 14 different toppings? D) A standard poker hand consists of five cards dealt from 52. How many different poker hands are possible?
