Problem 50
Question
Find the number of distinguishable permutations of the group of letters. \(\mathbf{M}, \mathbf{I}, \mathbf{S}, \mathbf{S}, \mathbf{I}, \mathbf{S}, \mathbf{S}, \mathbf{I}, \mathbf{P}, \mathbf{P}, \mathbf{I}\)
Step-by-Step Solution
Verified Answer
The number of distinguishable permutations of the given group of letters is \(11!/(1! * 4! * 4! * 2!)\).
1Step 1: 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).
2Step 2: 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.
3Step 3: 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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