Unformatted text preview: lem would be easier. There would be 9 ways to select
the ﬁrst player, and for each of those, 8 ways to select the second player, 7 ways to select the third,
6 ways to select the fourth, and 5 ways to select the ﬁfth. Thus, there are 9 · 8 · 7 · 6 · 5 ways to select
lineups in a given order. Since there are 5! ways to order a given lineup, each lineup would appear
5! times in a list of all possible lineups with all possible orders. So, by the principle of over counting,
the number of distinct lineups, with the order of a lineup not mattering, is 9 · 8 · 7 · 6 · 5/5! = 126. 4 “ILLINI,” pronounced “ill LIE nigh,” is the nickname for the students and others at the University of Illinois. 1.3. CALCULATING THE SIZE OF VARIOUS SETS 15 Example 1.3.6 How many binary sequences of length 9 have (exactly) 5 ones? One such sequence
is 110110010.
Solution: If the one’s were distinct and the zero’s were distinct, there would be 9! choices. But
there are 5! ways to order the one’s, and for each of those, 4! ways to order the zero’s, so there are
5! · 4! orderings with labels for each ordering with...
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 Spring '08
 Zahrn
 The Land

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