Jumble Many newspapers carry “Jumble,” a puzzle in which the r...

Question

Jumble Many newspapers carry “Jumble,” a puzzle in which the reader must unscramble letters to form words. The letters MHRHTY were included in newspapers on the day this exercise was written. How many ways can those letters be arranged? Identify the correct unscrambling, then determine the probability of getting that result by randomly selecting one arrangement of the given letters.

Accepted Answer

Welcome back, everyone. A popular word game presents players with a scrambled set of letters V E S E L L. Number one, how many ways can these letters be arranged? And number 2, if one arrangement is randomly selected, determine the probability that it forms incorrectly spelled word. Let's begin with the first part of the problem. We're going to visualize the problem by drawing 6 boxes because we have 6 letters given to us in the problem. So 12345, and 6 boxes. If we want to identify the total number of arrangements. That would be 6 for the first box because we have 6 available letters, followed by 5 remaining letters for the second box, 4 remaining letters for the 3rd box, and so on. 3. Followed by 2, and finally 1 or the 6 box. So now we're using the multiplication rule, we can say that this is 6 multiplied by 5, or 32, and 1, which is simply 6 factorial, right? However, we want to avoid overcounting, for example, If we interchange the two identical L letters, this doesn't form a new word, right, which is also similar to the two E letters. We have to take into account the fact that we have two L letters. In two exactly same E letters. So now how do we determine the actual total number of arrangements? Well, we have to take our 6 factorial and divide. By the factorial. Of the number of times each letter is repeated, for example, for 1, we have 2 as the number of repetitions, right? So we have to take 2 factorial and also we have to divide by another 2 factorial. Because E is also repeated 2 times. So dividing 2 times essentially means that we're dividing by the product of these factorials, right? And this gives us the total number of ways equal to 6 factorial. Divided by 2 factorial. And divided again by 2 factorial, which is equal to 180 ways, that would be our first answer. For the second part of the problem we want to answer the following. If one arrangement is randomly selected out of 180, determine the probability that it forms a correctly spelled word. So now how many words can we get from this set of letters that would be correctly spelled? Well, there is only one word, and that's levels. Right, so we want to identify the probability of a correctly spelled word. Let's suppose that this represents an event A. So P of A is equal to the number of favorable outcomes, that is 1 because we only have one possible correctly spelled word divided by the total number of outcomes, which is 180. And that will be our probability 1 divided by 180. That would be it. Thank you for watching.

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Key Concepts

Permutations

Factorial

Probability

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