In Ch. 4 of the Word Translations guide, I am getting confused on combinations vs. permutations.
The first practice problem asks: how many different ways can the letters in the word "LEVEL" be arranged?
Why is the solution 5!/(2!*2!)? It says the letters L and E have each been repeated twice. Seems like L has been repeated three times.
The second problem asks how many different boxes of chocolates can be made if there are five different types of chocolate, and each box has two different types of chocolate.
I'm struggling with the approach to this problem. Seems like there are 20 different possible combinations (5 for the first chocolate, and then 4 for the second).
I know you're supposed to set up an anagram model, but it's not making sense to me. Can someone break this down even more than the guide does?
Thanks
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- JonathanSchneider
- ManhattanGMAT Staff
- Posts: 370
- Joined: Sun Oct 26, 2008 3:40 pm
This is an area that a lot of folks struggle with. As to problem #1, notice that the first L and the second L are the same letter. Thus, writing them out with the first L starting off the word LEVEL is the same as if the second L were to start the word LEVEL. Although the letters are switched, we get the same word. This is the case for every arrangement, because we always have two L's. As a result, we need to divide by 2. That's what that 2! in the denominator is for. The same goes for the E's, because there are two of those.
For problem #2, you are right that it seems as though 20 would be correct. However, you are treating the problem as a permutation: finding the number of ways that we can order the chocolates. In fact, you must recognize that you are counting each possibility twice! Imagine you have a box with a marshmallow and an almond chocolate, and another box with an almond and a marshmallow chocolate. These are the same, so you cannot count them twice. Hence the answer comes out to half of the 20 you had predicted.
In general, the Anagram Model accounts for both combinations and permutations, which is why we promote it so much : )
For problem #2, you are right that it seems as though 20 would be correct. However, you are treating the problem as a permutation: finding the number of ways that we can order the chocolates. In fact, you must recognize that you are counting each possibility twice! Imagine you have a box with a marshmallow and an almond chocolate, and another box with an almond and a marshmallow chocolate. These are the same, so you cannot count them twice. Hence the answer comes out to half of the 20 you had predicted.
In general, the Anagram Model accounts for both combinations and permutations, which is why we promote it so much : )
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