Chapter 10 Counting Techniques

Miscellaneous Counting Methods Section 10.4

Permutations and Combinations Three cards are dealt in succession and without replacement from a standard deck of 52 cards. In how many different orders can the cards be dealt? How many different 3-card hands are possible? 52 nPr 3 = 52 x 51 x 50 = 132,600 52 x 51 x 50 3 x 2 x 1 52 nCr 3 = = 22,100

Permutations and Combinations The playbook for the quarterback of the Dallas Cowboys contains 50 plays. In how many different ways could the quarterback select 3 plays to use in succession in the next three downs? In how many different ways could he select a set of 3 plays to study? 50 x 50 x 50 = 125,000 50 x 49 x 48 3 x 2 x 1 50 nCr 3 = = 19,600

Permutations and Combinations Peter must select three electives from a group of 7 courses. In how many ways can Peter do this? If all 7 courses are available each of the 4 hours from 8 AM to noon, from how many different schedules (hours and what course at each hour) can Peter choose? 7 x 6 x 5 3 x 2 x 1 7 nCr 3 = = 35 7 nPr 4 = 7 x 6 x 5 x 4= 840

Distinct Arrangements The total number of distinct arrangements of n objects, where p objects are identical, q objects are identical, r objects are identical, and so on, is given by

Distinct Arrangements How many different arrangements can be made with the letters in the word TALLAHASSEE? with the letters in the word MISSISSIPPI? 11! 3!2!2!2! 39,916,800 6x2x2x2 39,916,800 48 = = = 831,600 11! 4!4!2! 39,916,800 24x24x2 39,916,800 1152 = = = 34,650 END