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Chapter 9: Problem 50

Find the number of distinguishable permutations of the group of letters. \(\mathbf{M}, \mathbf{I}, \mathbf{S}, \mathbf{S}, \mathbf{I}, \mathbf{S}, \mathbf{S}, \mathbf{I}, \mathbf{P}, \mathbf{P}, \mathbf{I}\)

Short Answer

Expert verified
The number of distinguishable permutations of the given group of letters is \(11!/(1! * 4! * 4! * 2!)\).

Step by step solution

01

Identify the Elements and their Frequencies

The given group of letters is \(M,I,S,S,I,S,S,I,P,P,I\) and the frequencies of these elements are: 'M' (1 time), 'I' (4 times), 'S' (4 times) and 'P' (2 times).
02

Total Number of Elements

Add up the frequencies of all the elements to find the total number of elements, n. So here, the total number of letters is 1 + 4 + 4 + 2 = 11.
03

Calculate the Number of Distinguishable Permutations

To find the number of distinguishable permutations, use the formula for permutations of a multiset which is \( n!/(r1! * r2! * ... * rn!)\) , with n being the total number of elements and r1, r2, ..., rn being the number of occurrences of each individual element. Thus, it is \(11!/(1! * 4! * 4! * 2!)\).

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