Find the number of distinguishable arrangements of the letters of each word.

Question

1) C O M M I S S I O N 

 

2) H E E B I E - J E E B I E S

Answer 1

This problem can be understood through permutations, while also accounting for the fact that identical letters cannot be distinguished from one another.

 

In general, if you have n unique symbols, and you want to calculate the number of permutations, there are n! (n factorial) ways to permute these symbols.  Essentially, you have n choices for the first position, (n-1) choices for second position, (n-2) choices for the 3rd, etc., which you multiply all together, and that is the same as the factorial definition.

 

In the case of duplicated symbols, you have what is referred to as a multi-set instead, where instead of ignoring duplicate elements, you have a count of anything that occurs more than once.

 

In the case of COMMISSION, you have the multi-set { 1 * C, 1 * N, 2 * O, 2 * I, 2 * M, 2 * S }.  There are 10 total symbols, but there are not 10! permutations, because some of these cannot be distinguished from one another due to identical letters switching among their shared positions.  Therefore you have to divide by the ways in which identical letters can be permuted among their shared positions, which is also a factorial of the number of times that letter occurs.

 

So with n total symbols, but of a multi-set where the symbols have repetition number k_1, k_2, k_3, etc., the total number of permutations will be

 

n! / ( k_1! * k_2! * k_3! * ...)

 

In the case of COMMISSION, you will have

10! / ( 1! * 1! * 2! * 2! * 2! * 2!) = 226800

 

In the case of HEEBIEJEEBIES, which is the multi-set { 1 * H, 1 * S, 2 * I, 2 * B, 6 * E } you will have

12! / ( 1! * 1! * 2! * 2! * 6! ) = 166320