Permutations with Identical Terms
Often, you will deal with permutations in which some items are identical. you can develop a general formula for permutations in which some items are identical:
This formula represents \(P\) as the number of permuations, for a set of \(n\) items of which \(a\) are identical.
The number of permutations of a set of \(n\) objects containing \(a\) identical objects of one kind, \(b\) identical objects of a second kind, \(c\) identical objects of a third kind, and so on is:
There are 11 Total letters. Outlined below are the individual amounts for each letter:
- \(\text{M} = 1\)
- \(\text{I} = 4\)
- \(\text{S} = 4\)
- \(\text{P} = 2\)
Next, we can substitute these values into the equation and solve:
\(P = \cfrac{11!} {(1!) (4!) (4!) (2!)}\)
\(P = 34,650\)
Therefore, we can determine there are 34,650 different ways the word MISSISSIPPI can be arranged.
There are 7 Total books. Outlined below are the subjects with their corresponding number of identical books:
- \(\text{Math} = 3\)
- \(\text{History} = 2\)
Next, we can substitute these values into the equation and solve:
\(P = \cfrac{7!}{(3!) (2!)}\)
\(P = 420\)
Therefore, we can determine the books can be arranged in 420 different ways.