Which expression represents the number of distinct arrangements of n distinct objects (permutations)?

Which expression represents the number of distinct arrangements of n distinct objects (permutations)?

• A. 2^n
• B. n
• C. n!
• D. C(n, k)

Answer: (C)

When you want to arrange all n distinct objects in a sequence, the order in which you place them matters, and each step reduces the number of choices by one. For the first position you have n options, for the second you have n−1, then n−2, all the way down to 1. Multiply these together and you get n!, the number of distinct permutations of all n objects. For a quick check, with three objects there are 3! = 6 different orders (ABC, ACB, BAC, BCA, CAB, CBA), which matches the idea above. The other expressions don’t count full arrangements of all items. 2^n counts all subsets or binary choices for each object, not the different orders. The lone n counts only selecting one object or a single choice, not arranging all of them. C(n, k) counts ways to choose k objects without regard to order, which is different from arranging every object. Thus the expression for the number of distinct arrangements is n!.

When you want to arrange all n distinct objects in a sequence, the order in which you place them matters, and each step reduces the number of choices by one. For the first position you have n options, for the second you have n−1, then n−2, all the way down to 1. Multiply these together and you get n!, the number of distinct permutations of all n objects. For a quick check, with three objects there are 3! = 6 different orders (ABC, ACB, BAC, BCA, CAB, CBA), which matches the idea above.

The other expressions don’t count full arrangements of all items. 2^n counts all subsets or binary choices for each object, not the different orders. The lone n counts only selecting one object or a single choice, not arranging all of them. C(n, k) counts ways to choose k objects without regard to order, which is different from arranging every object. Thus the expression for the number of distinct arrangements is n!.