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- How many ways can the letters of the word "ARRANGE" be arranged such that the two A's are never together? Option: 1 264
How many ways can the letters of the word "ARRANGE" be arranged such that the two A's are never together?
2640
5689
4980
7800
Answers (1)
To calculate the number of ways the letters of the word "ARRANGE" can be arranged such that the two A's are never together, we can consider the total number of arrangements and subtract the number of arrangements where the two A's are always together.
First, let's calculate the total number of arrangements of the letters in "ARRANGE". The word has a total of 7 letters, so the total number of arrangements is .
Next, let's calculate the number of arrangements where the two A's are always together. We can treat the two A's as a single entity. This means we have "(AA)" and the remaining entities are "R, R, N, G, E". We can arrange these 5 entities in ways. However, within this arrangement, the two R's repeat, so we need to divide by
to account for the repetition.
Therefore, the number of arrangements where the two A's are always together is .
Finally, we subtract the number of arrangements where the two A's are always together from the total number of arrangements to get the number of arrangements where the two A's are never together:
.
Calculating this expression, we have:
.
Thus, there are 4980 ways to arrange the letters of the word "ARRANGE" such that the two A's are never together.
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