Counting Permutations When Indistinguishable Objects May Exist

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Counting Permutations When Indistinguishable Objects May Exist

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Counting Permutations When Indistinguishable Objects May Exist

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How many rows , each one consisting of 3 A’s 1 B, and 4 C’s are there?

(Here are some such rows:

BACCCAAC

ABCACACC

CCCCAAAB

Etc.)

Answer: (3+1+4)! / (3!1!4!).

In general:

k distinct types of objects are given, where k is a positive integer.

Positive integers ni for i=1,…k are given. k is

a given positive integer. How many rows are there, each one including ni objects of type i

for i=1, …, k and no other objects?

Answer: Let n=n1 + … +nk. Then the number of rows is n!/(n1! … nk!).