Does any letter in our group occur more than once?
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We want to find the number of distinguishable permutations of the group of letters.
I, N, T, U, I,T, I, O, N
First, let's split the letters into groups that contain the same letter.
ccc
I, I, I & & N, N
T, T & O & U
The letters are divided into five groups. Let's count the letters in each group.
Letter
Number of Copies of the Letter
I
n_1= 3
N
n_2= 2
T
n_3= 2
O
n_4 = 1
U
n_5= 1
When we want to find the number of distinguishable permutations of a set with n elements, in which there are n_1 elements of one kind, n_2 of a second, and so on with n = n_1+...+n_k, we can use the formula for permutations with repetitions.
n!/n_1!* n_2!* ... * n_k!
In our case n = 9, n_1 = 3, n_2 = 2, n_3 = 2, n_4=1, and n_5 = 1. Let's substitute these values into the formula for the number of distinguishable permutations of our set!
