1091 11.6 Basics of Counting Theory mystery-biography-textbook biography-textbook-mystery mystery-textbook-biography textbook-biography-mystery biography-mystery-textbook textbook-mystery-biography The list shows 6 different arrangements of 3 books but only one set of 3 books. A subset of items selected without regard to order is a combination. The number of combinations of 5 things taken 3 at a time is written C15, 32 or 5C3. To evaluate C15, 32, start with the 5 # 4 # 3 permutations of 5 things taken 3 at a time. Because order does not matter, and each subset of 3 items from the set of 5 items can have its elements rearranged in 3 # 2 # 1 = 3! ways, we find C15, 32 by dividing the number of permutations by 3!. C15, 32 = 5 # 4 # 3 3! = 5 # 4 # 3 3 # 2 # 1 = 10 The teacher can choose 3 books for the book sale in 10 ways. Generalizing this discussion gives the following formula for the number of combinations of n elements taken r at a time. C1n, r2 = P1n, r2 r! An alternative version of this formula is found as follows. C1n, r2 = P1n, r2 r! = n! 1n - r2! # 1 r! = n! 1n - r2! r! A seating arrangement is a permutation. EXAMPLE 6 Using the Permutations Formula In how many ways can 6 students be seated in a row of 6 desks? SOLUTION P16, 62 = 6! 16 - 62! Permutations formula with n = r = 6 = 6! 0! Subtract; 0! = 1 = 6 # 5 # 4 # 3 # 2 # 1 Definition of 6! = 720 ways Multiply. S Now Try Exercise 45. Combinations In Example 3 we saw that there are 60 ways in which a teacher can arrange 3 of 5 different books in a row. That is, there are 60 permutations of 5 things taken 3 at a time. Suppose now that the teacher does not wish to arrange the books in a row but rather wishes to choose, without regard to order, any 3 of the 5 books to donate to a book sale. In how many ways can the teacher do this? The number 60 counts all possible arrangements of 3 books chosen from 5. The following 6 arrangements, however, would all lead to the same set of 3 books being given to the book sale. NOTE The formula for C1n, r2 given above is equivalent to the binomial coefficient formula, symbolized An r B, studied earlier in the chapter.
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