![]() Thus we must divide our answer of \(6!\) by \(3!\) to account for the groupings of six that become identical. These six arrangements would be seen as the same if the Es were no longer distinct: But the three Es can be rearranged \(3! = 6\) different ways within any one particular arrangement of letters. How many ways are there to rearrange the letters of the word CHEESE?Īnswer: If the three Es are distinct – written E 1, E 2, and E 3, say – then there are \(6!\) ways to rearrange the letters CHE 1E 2S E 3. ![]() This agrees with the answer \(6 \times 5 \times 4 \times 3 \times 1\). We must alter our answer by a factor of two and so the number of arrangements of the word HOUSES is: ![]() īut notice, if the Ss are no longer distinguishable, then pairs in this list of answers “collapse” to give the same arrangement. There are \(6!\) ways to rearrange the letters HOUS 1ES 2. What if it didn’t have repeat letters? Suppose the Ss were distinguishable, written, say, as S 1 and S 2. It uses a useful strategy in problem-solving:ĬAN I CONVERT THE PROBLEM TO SOMETHING I HAVE SOLVED BEFORE?ĪPPROACH 2: The problem with HOUSES is that it has repeat letters. (Try it!)Īllow me to suggest a second approach that does generalize nicely. But it is not very helpful, however, for words (names) like ABBA and MISSISSIPPI. This approach works well, at least for HOUSES. There are \(3\) ways to accomplish this task.Īs there are only two slots left, there is only \(1\) way to accomplish this task.īy the multiplication principle there are \(6 \times 5 \times 4 \times 3 \times 1\) arrangements of the letters HOUSES. With the H and O in place, there are \(4\) ways to accomplish this task. With the H in position, there are \(5\) ways to accomplish this task. There are \(6\) ways to accomplish this task. Here’s one way to think of this problem …ĪPPROACH 1: Think of this as a five-stage multitask. Does listing all the possibilities seem fun? In how many ways can one arrange the letters HOUSES? Do you think there might be another way to tackle this problem? What if my name were BOB? How might we handle repeated letters?Įxercise 11: How many ways are there to arrange the letters BOB? Assume the Bs are indistinguishable?In how many ways can one rearrange the letters of ABBA?Īnswer exercise 11. We can rearrange the letters of JIM and of JAMES and of other words. For example, there are \(6 \times 5 \times 4 \times 3 \times 2 \times 1 = 6!\) ways to arrange the letters of BOVINE. The factorials arise as answers to word rearrangement problems. What is the largest factorial your calculator can handle? ON A CALCULATOR the factorial feature is usually hidden under the some PROBABILITY menu. What is the first factorial larger than a million? A billion? (Does the definition given make sense for \(1!\) ? Does it worry you I write the products from \(N\) down to \(1\) instead of from \(1\) up to \(N\)?) These factorial numbers grow very large very quickly: When playing with these problems it is clear that the following definition is needed:ĭefinition: For a given counting number \(N\), the product of integers from \(1\) to \(N\) is called factorial and is denoted \(N!\). But if we think of this as a counting problem with five tasks – place a first letter, place a second letter, and so on – we see that there must be \(5 \times 4 \times 3 \times 2 \times 1 = 120\) arrangements of the letters of JAMES.Įxercise 9: In how many ways can one arrange the letters BOVINE ?Įxercise 10: In how many ways can one arrange the letters FACETIOUSLY?(What is unusual about the vowels of this word? Find another word in the English language with this property!) In how many ways can one arrange the letters of my proper first name?Īnswer: The brute force method wouldn’t be fun this time. ![]() There is only 1 way to complete this task (once slots one and two are filled).īy the multiplication principle, there are \(3 \times 2 \times 1 = 6\) ways to complete this job.Īctually, my formal name is JAMES. The third task is to fill the third slot. (Once the first slot is filled, there are only two choices of letter to use for the second slot.) The second task is to fill the second slot. The first task is to fill the first slot with a letter. (This is called the brute force method!)Īnswer 2: Use the multiplication principle: We have three slots to fill: In how many ways can I arrange the letters of my name?Īnswer 1: We could just list the ways.
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