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Unformatted text preview: many strings of length r can be formed from the uppercase letters of the English alphabet?
Solution: The number of such strings is 26r, which is the number of r‐permutations of a set with 26 elements. 24 Combinations with Repetition
Example: How many ways are there to select five bills from a box containing at least five of each of the following denominations: $1, $2, $5, $10, $20, $50, and $100? Solution: Place the selected bills in the appropriate position of a cash box illustrated below: continued 25 Combinations with Repetition
Some possible ways of placing the five bills: The number of ways to select five bills corresponds to the number of ways to arrange six bars and five stars in a row. This is the number of unordered selections of 5 objects from a set of 11. Hence, there are ways to choose five bills with seven types of bills.
26 Combinations with Repetition
Theorem 2: The number 0f r‐combinations from a set with n
elements when repetition of elements is allowed is
C(n + r – 1,r) = C(n + r – 1, n –1).
Proof: Each r‐combination of a set with n elements with repetition allowed can be represented by a list of n –1 bars and r
stars. The bars mark the n cells containing a star for each time the ith element of the set occurs in the combination.
The number of such lists is C(n + r – 1, r), because each list is a choice of the r positions to place the stars, from the total of n + r – 1 positions to place the stars and the bars. This is also equal to C(n + r – 1, n –1), which is the number of ways to place the n –1 bars.
27 Combinations with Repetition
Example: How many solutions does the equation
x1 + x2 + x3 = 11
have, where x1 , x2 and x3 are nonnegative integers?
Solution: Each solution corresponds to a way to select 11
items from a set with three elements; x1 elements of type one, x2 of type two, and x3 of type three. By Theorem 2 it follows that there are solutions. 28 Combinations with Repetition
Example: Suppose that a cookie shop has four different kinds of cookies. How many different ways can s...
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This document was uploaded on 03/29/2014 for the course COT 3100h at University of Central Florida.
 Spring '08
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