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What is Permutations and combinations?

Ways of counting the number of cases when some items are chosen from a set. A permutation distinguishes the order of the chosen items: the number of ways to choose and arrange r of n items is nPr = n!/(n − r)!. A combination ignores the order: nCr = n!/(r!(n − r)!) = nPr/r!. With repetition allowed, there are nʳ arrangements. They are the basis of probability calculations and are also used to estimate the total number of PINs or the number of test combinations.

Japanese name: 順列と組合せ

Permutations and combinations appears in 4 practice questions across 1 topic, including Probability, Statistics and Queueing.

Field: Probability, Statistics and Queueing

Try it: questions about Permutations and combinations

Three committee members are chosen from 8 people. If the members have no distinct roles, which of the following is the number of possible choices?

  1. A56
  2. B336
  3. C512
Show answer and explanation

Answer: A. 56

Since the members have no distinct roles, the order of selection does not matter, so the number of combinations 8C3 is needed. 8C3 = (8 × 7 × 6) ÷ (3 × 2 × 1) = 56. It can also be seen as the number of permutations 8P3 = 336, which distinguishes the order, divided by 3! = 6, the number of ways to order the same three people.

A chair, a vice-chair and a secretary are chosen, one each, from 5 people. If no one may hold two or more roles, which of the following is the number of possible choices?

  1. A10
  2. B60
  3. C125
Show answer and explanation

Answer: B. 60

Because the roles differ, the number of permutations 5P3 is needed, which also distinguishes who is assigned to which role. There are 5 choices for the chair, 4 of the remaining people for the vice-chair and 3 of those still remaining for the secretary, so 5 × 4 × 3 = 60. Multiplying the number of combinations without roles, 5C3 = 10, by the 3! = 6 ways of assigning the roles to the three people also gives 60.

Permutations and combinations in the explanations

Questions that mention Permutations and combinations (4)

  1. Three committee members are chosen from 8 people. If the members have no distinct roles, which of the following is the number of possible choices?
  2. A chair, a vice-chair and a secretary are chosen, one each, from 5 people. If no one may hold two or more roles, which of the following is the number of possible choices?
  3. A 4-digit PIN is made by arranging four of the digits 0 to 9. If the same digit may be used any number of times and the first digit may be 0, which of the following is the number of possible PINs?
  4. In a town whose streets form a square grid, you go from intersection A to intersection B, which is 4 blocks east and 3 blocks north of A, by a shortest route that moves only east or north. Which of the following is the number of routes?

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Related terms

Bayes' theorem Binomial distribution Conditional probability Confidence interval Correlation coefficient Expected value Exponential distribution Hypothesis testing Median Normal distribution

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