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  • OBJECTIVES

    Counting PrinciplesLearn to use the Fundamental Counting Principle. Learn to use the formula for permutations. Learn to use the formula for combinations. Learn to use the formula for distinguishable permutations.SECTION 11.61234

  • FUNDAMENTAL COUNTING PRINCIPLEIf a first choice can be made in p different ways, a second choice can be made in q different ways, a third choice can be made in r different ways, and so on, then the sequence of choices can be made in p q r different ways.

  • EXAMPLE 3Counting Possible Social Security NumbersSocial Security numbers have the format NNN-NN-NNNN, where each N must be one of the integers 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Assuming that there are no other restrictions, how many such numbers are possible? 101010101010101010 = 109There are 1,000,000,000 possible Social Security numbers.

  • DEFINITION OF PERMUTATIONA permutation is an arrangement of n distinct objects in a fixed order in which no object is used more than once. The specific order is important: Each different ordering of the same objects is a different permutation.

  • NUMBER OF PERMUTATIONS OF n OBJECTSThe number of permutations of n distinct objects isThat is, n distinct objects can be arranged in n! different ways.

  • PERMUTATIONS OF n OBJECTS TAKEN r AT A TIMEThe number of permutations of n distinct objects taken r at a time is denoted by P(n, r), where

  • EXAMPLE 6Using the Permutation FormulaUse the formula for P(n, r) to evaluate each expression.a. P(7, 3) b. P(6, 0)

  • DEFINITION OF A COMBINATION OF n DISTINCT OBJECTS TAKEN r AT A TIMEWhen r objects are chosen from n distinct objects without regard to order, we call the set of r objects a combination of n objects taken r at a time.The symbol C(n, r) denotes the total number of combinations of n objects taken r at a time.

  • THE NUMBER OF COMBINATIONS OF n DISTINCT OBJECTS TAKEN r AT A TIMEThe number of combinations of n distinct objects taken r at a time is:

  • EXAMPLE 9Choosing Pizza ToppingsHow many different ways can five pizza toppings be chosen from the following choices: pepperoni, onions, mushrooms, green peppers, olives, tomatoes, mozzarella, and anchovies?There are 56 ways to select 5 of 8 pizza toppings.

  • DISTINGUISHABLE PERMUTATIONSThe number of distinguishable permutations of n objects of which n1 are of one kind, n2 are of second kind, , and nk are of a kth kind iswhere n1 + n1 + + nk = n.

  • EXAMPLE 10Distributing GiftsIn how many ways can nine gifts be distributed among three children if each child receives three gifts?There are 1680 ways the nine gifts can be distributed among the three children.

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