Understanding the Permutations of the Word Committee and Beyond

Introduction to Permutations in a Multiset Context

In addressing the question of how many permutations there are in the word committee, we delve into the concept of permutations within a multiset. A multiset is a set with repeated elements, and the calculation of the total number of distinct permutations involves a specific formula. Let's explore the process step-by-step.

Calculating Permutations of the Word ' committee '

The word committee consists of 9 distinct letters, with repetitions as follows:

c: 1 o: 1 m: 1 i: 2 t: 1 e: 2

The total number of letters, n, is 9. For a multiset where some items may be repeated, the formula for calculating permutations is:

[ text{Permutations} frac{n!}{n_1! times n_2! times cdots times n_k!} ]

Here, n is the total number of letters, and n_1, n_2, dots, n_k are the frequencies of the repeated letters.

Applying the Formula to ' committee '

Substituting the values into the formula:

[ text{Permutations} frac{9!}{1! times 1! times 1! times 2! times 1! times 2!} ]

Calculating each factorial:

9! 362880 1! 1 2! 2

Hence, the number of permutations is:

[ text{Permutations} frac{362880}{1 times 1 times 1 times 2 times 1 times 2} frac{362880}{4} 90720 ]

Therefore, the number of permutations of the word committee is 90720.

Permutations of Other Multisets

Let's explore additional examples to solidify our understanding.

Permutations of a 10-Letter Word: COMMITTEES

The word COMMITTEES consists of 10 letters with repetitions:

C: 1 o: 1 M: 2 I: 2 T: 2 EE: 2

Using the formula:

[ text{Permutations} frac{10!}{2! times 2! times 2! times 2!} ]

Calculating each factorial:

10! 3628800 2! 2

To find the total permutations:

[ text{Permutations} frac{3628800}{2 times 2 times 2 times 2} frac{3628800}{16} 226800 ]

Hence, the number of distinct arrangements of COMMITTEES is 226800.

What if Letters Were Distinct?

Consider the word:

comMitTeE

With uppercase and lowercase letters treated as distinct, the total number of arrangements is:

[ text{Arrangements} 9! 362880 ]

Now, convert each uppercase letter to lowercase:

For the uppercase M, there is exactly one other arrangement where the M and m are switched, hence:

[ text{Distinct arrangements} frac{9!}{2} 181440 ]

Similarly, for the uppercase T and E:

[ text{Distinct arrangements} frac{9!}{2 times 2} 90720 ]

General Rule for Permutations in Multisets

When dealing with a multiset with:

n_1 letters of L_1 n_2 letters of L_2 ldots n_k letters of L_k

The number of distinct permutations is given by:

[ text{Permutations} frac{n_1 times n_2 times cdots times n_k!}{n_1! times n_2! times cdots times n_k!} ]

Conclusion

We have explored the process of determining the number of permutations for the word committee and other examples of multisets. The understanding of permutations in multisets is crucial for various combinatorial problems in mathematics and computer science.