A distinguishable permutation refers to the number of unique arrangements of a set of objects where some objects may be identical. In combinatorics, when we deal with permutations that include repetitions, we need to account for the indistinguishable items to avoid overcounting.
For example, consider the letters in the word ‘BALLOON’. If we simply calculate the total arrangements of the 7 letters, we would get 7! (factorial of 7). However, we have to adjust this number to reflect that the letters ‘L’ and ‘O’ appear more than once. The formula to calculate distinguishable permutations is:
P = n! / (n1! * n2! * ... * nk!)Here, n is the total number of items to arrange, and n1, n2, …, nk is the factorial of each group’s identical items.
In the case of ‘BALLOON’:
- Total letters (n): 7
- Repetitions: 1 B, 1 A, 2 L’s, 2 O’s, 1 N
Applying the formula:
P = 7! / (1! * 1! * 2! * 2! * 1!)This results in the number of distinguishable permutations of ‘BALLOON’, effectively providing the unique arrangements possible. Understanding distinguishable permutations is crucial in probability and combinatorial problems where you have groups of similar items.