# Probability - Permutations and Combinations

by BJ Hutchinson
(Indianapolis, In)

Probability - Word Problem

If eleven people enter room and six chairs are available, how many ways can you fill the six chairs?

### Comments for Probability - Permutations and Combinations

 Feb 17, 2013 Word Problem by: Staff Answer Part I Your problem does state how the order of the people who sit in the chairs should be taken into account. Does the order of the people in the chairs make a difference? Is the arrangement of the people sitting in the chairs {1, 2, 3, 4, 5, 6} the same as {6, 5, 4, 3, 2, 1}? If the order (the sequential arrangement of the individual people sitting in the chairs) does not matter, it called a Combination. A Combination means that a seating arrangement of {1, 2, 3, 4, 5, 6} is the same as {6, 5, 4, 3, 2, 1} is the same as {4, 5, 6, 3, 2, 1}, and so on. These three seating arrangements are counted a one combination. All three seating arrangements have exactly the same numbers, regardless of the order. If the order (the sequential seating arrangement of the different people) is important, it called a Permutation. The permutation is a list of all possible seating arrangements. A Permutation means that a seating arrangement of {1, 2, 3, 4, 5, 6} is the NOT same as {6, 5, 4, 3, 2, 1} is the NOT same as {4, 5, 6, 3, 2, 1}, and so on. These three seating arrangements are counted a three permutations. All three seating arrangements may have the same numbers, but the order is different. Permutations For your problem, if the order in which people are seated does matter: 0 ≤ r ≤ n. n = how many different people enter the room ( = 11) r = number of chairs available (= 6) Order is important Repetition is not allowed -----------------------------

 Feb 17, 2013 Word Problem by: Staff ----------------------------- Part II If seating order does matter, there are : 332,640 possible seating arrangements Combinations For your problem, if order in which people are seated does not matter: Note that the formula for combinations is almost the same as the formula for permutations. The only difference is the number of permutations nPr is divided by r! to eliminate the duplicates. 0 ≤ r ≤ n. n = how many different people enter the room ( = 11) r = number of chairs available (= 6) Order is not important Repetition is not allowed If seating order does not matter, there are : 462 possible seating arrangements Thanks for writing. Staff www.solving-math-problems.com

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