Page 207 - Math Course 3 (Book 2)
P. 207

Permutations and Combinations





                 MO. 12 - L6b                                                 n!             Definition of
                                                                      C  =
                                                                     n r   (n – r)!r!        combination
                         Using Combinations                                     22!          n = 22, r = 4
                                                                     C  =
                                                                   22 4      (22 – 4)!4!
                                                                            22!
                                                                             =  18!4!        22 – 4 = 18
                            Vocabulary A-Z                                                1

                            Let us learn some vocabulary                     =  22 • 21 • 20 • 19 • 18 !  Divide by the GCF,
                                                                                             18!.
                                                                                 18! 4!
                                                                               1
                                                                          175, 560
                                                                             =                     or 7315  Simplify.
                combination                                                  24
                An arrangement or listing in which order is not
                important is called a combination.                    Answer       There are 7315 ways to pull 4
                                                                                     coins out of a bag of 22.
                   If you are choosing 2 salad ingredients from
                  a list of 10, the order in which you choose the   There are two questions to consider.
                          ingredients does not matter.
                                                                   • How many ways can 2 pennies be pulled from 10?
                                                                   • How many ways can 2 nickels be pulled from 6?
                Key Concept                                        Using the Fundamental Counting Principle, the


                                                                   answer can be determined with the product of the
                                                                   two combinations.
                Combination                                             ways to choose     ways to choose 2

                                                                        2 pennies out      nickels out
                Words     The number of combinations of n               of 10              of 6
                          objects taken r at a time is the quotient
                          of n! and (n – r)!r!.
                                                                            ( C )       ●       ( C )
                                                                                                   2
                                                                            10
                                                                               2
                                                                                                 6
                Symbols    C  =    n!
                          n r   (n – r)!r!                                        10!        6!    Definition of
                                                                  ( C )( C ) =                      •
                                                                   10 2 6 2    (10–2)!2!   (6–2)!2!  combination
                                                                                10!    6!
                                                                                       =             •  4! 2!  Simplify.
                                                                               8! 2!
                             Let’s Begin                                                           Divide the first
                                                                                                   term by its
                                                                               10 • 9
                                                                                       =               •  6 • 5  GCF, 8!, and the
                                                                                 2!      2!        second term by
                                                                                                   its GCF, 4!.
               Combinations and Probability
                                                                                       = 675       Simplify.
                   Example                                         There are 675 ways to choose this particular
                                                                   combination out of 7315 possible combinations.

                                                                                675
                MONEY                                             P(2 pennies,   = 7315 ←number of favorable outcomes
                                                                  2 nickels)
                                                                                    ←number of possible outcomes
                Diane has a bag full of coins. There are 10 pennies,             135
                6 nickels, 4 dimes, and 2 quarters in the bag. How            =  1463       Simplify
                many different ways can Diane pull four coins out
                of the bag?
                                                                                   The probability that Diane will
                The order in which the coins are chosen does not      Answer        select two pennies and two
                matter, so we must find the number of combina-                     nickels is            , or about 9%.
                                                                                            135
                tions of 22 coins taken 4 at a time.                                        1463

                                                                                                                  199
   202   203   204   205   206   207   208   209   210   211   212