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Chapter 11: Problem 83

Explain the difference between permutations and combinations.

Short Answer

Expert verified
Permutations consider order, combinations do not.

Step by step solution

01

Define Permutations

Permutations refer to the arrangement of items in a specific order. The order in which the items are arranged is important. For permutations of a set of n items taken r at a time, the formula is given by: \[ P(n, r) = \frac{n!}{(n-r)!} \] where '!' denotes factorial.
02

Define Combinations

Combinations refer to the selection of items from a set where the order does not matter. For combinations of a set of n items taken r at a time, the formula is given by: \[ C(n, r) = \frac{n!}{r!(n-r)!} \] where '!' denotes factorial.
03

Compare Permutations and Combinations

The key difference is that in permutations, the order of items is important, whereas in combinations, the order of items does not matter. This means that permutations count different sequences of the same items as unique, while combinations count them as the same.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Permutations: Importance of Order
In mathematics, permutations are used when the order of items truly matters. For instance, think of arranging books on a shelf. If you have 5 books and you want to arrange 3 of them, the order you pick can drastically change the outcome. The formula to find permutations of a set of n items arranged r at a time is given by: \[ P(n, r) = \frac{n!}{(n-r)!} \] Here, the symbol '!' denotes factorial, which means the product of all positive integers up to that number. Thus, the order is crucial in permutations; changing the sequence of items yields a different permutation.
Combinations: Order Does Not Matter
Combinations are used when the order of selected items does not matter. Consider selecting a team of 3 members from a group of 5. No matter the order in which you pick the members, the team remains the same. This is what sets combinations apart from permutations. The formula to find combinations of a set of n items taken r at a time is: \[ C(n, r) = \frac{n!}{r!(n-r)!} \] Again, '!' represents factorial. So, in combinations, changing the order of selection does not create a new combination; it remains consistent irrespective of the sequence.
Factorials in Permutations and Combinations
Factorials are a key component in both permutations and combinations, playing a critical role in their calculations. Factorials are denoted by '!', and are defined as the product of an integer and all the integers below it. For example:
  • \[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]
When calculating permutations, we use factorials to account for all possible arrangements. For combinations, factorials help to remove redundancies where the order does not matter. Thus, understanding and using factorials is essential for mastering permutations and combinations.

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