5.5 Generalized Permutations and
Combinations
Permutations with Repetition
• Theorem 1: The number of r-permutations of a set of
n objects with repetition allowed is nr .
• Example 1: How many strings of length r can be
formed from the English alphabet?
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Combinations with Repetition
• Theorem 2: There are C(n+r-1, r) = C(n+r-1, n-1)
r-combinations from a set with n elements when
repetition of elements is allowed.
• Example 4: Suppose that a cookie shop has four
different kinds of cookies. How many different ways
can six cookies be chosen? Assume that only the
type of cookie, and not the individual cookies or the
order in which they are chosen, matters.
• Example 5: How many solutions does the equation
x1 + x2 + x3 = 11
have, where x1, x2, and x3 are nonnegative integers?
• HW: Example 3,(p.372)
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Combinations with Repetition
• Example 6: What is the value of k after the following
pseudocode has been executed?
k :=0
for i1 := 1 to n
for i2 := 1 to i1
:
for im := 1 to im-1
k :=k+1
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Combinations with Repetition
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Permutations with Indistinguishable Objects
• Some elements may be indistinguishable in counting
problems. When this is the case , care must be taken
to avoid counting things more than once. Consider
example 7.
• Example 7: How many different strings can be made
by reordering the letters of the word SUCCESS?
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Permutations with Indistinguishable Objects
• Theorem 3: The number of different permutations of
n objects, where there are n1 indistinguishable
objects of type 1, n2 indistinguishable objects of type
2,. . . , and nk indistinguishable objects of type k, is
n! /(n1! n2! ‥nk!)
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Distributing Objects into Boxes
• Distinguishable Objects and Distinguishable Boxes
• We first consider the case when distinguishable
objects are placed into distinguishable boxes.
Consider example 8 in which the objects are cards
and the boxes are hands of players.
• Theorem 4: The number of ways to distribute n
distinguishable objects into k distinguishable boxes
so that ni objects are placed into box i, i=1, 2, . . .,k,
equals n! /(n1! n2!‥nk!)
• Example 8: How many ways are there to distribute
hands of 5 cards to each of four players from the
standard deck of 52 cards?
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Distributing Objects into Boxes
• Indistinguishable Objects and Distinguishable Boxes
• example 9: How many ways are there to place 10
indistinguishable balls into eight distinguishable bins?
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Distributing Objects into Boxes
• Distinguishable Objects and Indistinguishable Boxes
• Example 10: How many ways are there to put four
different employees into three indistinguishable
offices, when each office can contain any number of
employees?
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Distributing Objects into Boxes
• Indistinguishable Objects and Indistinguishable Boxes
• Example 11: How many ways are there to pack six
copies of the same book into four identical boxes,
where a box can contain as many as six books?
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