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# What are some distinguishable permutations?

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Examples of distinguishable permutations often include letter order in words, such as the number of ways in which the letters in the word "Mississippi" can be uniquely arranged. Distinguishable permutations are also used for research, such as for diet or vaccination trials.

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To use the mathematical formula for distinguishable permutations in the word "Mississippi," the factorial function of the number of letters in the word symbolized by "!" must be divided by the factorial functions of each of the unique letters in the word. In this case the number of letters is 11. There are four instances of the letter "s," four instances of "i," two instances of "p" and one instance of "m" in "Mississippi," so the factorial functions are "4! 4! 2! 1!" Thus, the full formula for finding the distinguishable permutations in "Mississippi" is "11! / 4! 4! 2! 1!"

The first step in solving this problem is to open it: "[(11)(10)(9)(8)(7)(6)(5)(4)(3)(2)(1)] / [(4)(3)(2)(1)] [(4)(3)(2)(1)] [(2)(1)] [(1)]." The second step is to cancel, in this case remove, the 1, 2, 3, 4, 6 and 8 from the numerator and all the denominators to end up with "[(11)(10)(9)(7)(5)]." The final step is to solve the problem by multiplying. The final answer, 34,650, is the total number of unique, 11-letter ways in which the letters in the word "Mississippi" can be arranged.

The same process can be used to find out the number of combinations for certain types of clinical trials. For example, if three types of dog food were tested by a group of 15 dogs so that each type was tested by five dogs, the formula would be "15! / 5! 5! 5!" The answer, 756,756, is the total number of unique combinations that can be made in such a trial.

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