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- How many ways can the letters of the word "ARRANGE" be arranged such that the two R's are never together?Option: 1 20,640
How many ways can the letters of the word "ARRANGE" be arranged such that the two R's are never together?
20,640
56,600
66,400
37,800
Answers (1)
To calculate the number of ways the letters of the word "ARRANGE" can be arranged such that the two R's are never together, we can consider treating the two R's as distinct entities (R1 and R2).
The word "ARRANGE" has a total of 7 letters, but since the two R's are treated as distinct entities, we have a total of 8 entities to arrange (A, R1, R2, A, N, G, E).
We can arrange these 8 entities in 8! ways. However, within this arrangement, we need to consider the cases where the two R's are together.
To calculate the number of arrangements where the two R's are together, we can treat the two R's as a single entity. Then we have 7 entities to arrange (A, (R1R2), A, N, G, E). We can arrange these 7 entities in ways. However, within this arrangement, the two A's repeat, so we need to divide by
Therefore, the number of arrangements where the two R's are together is (7! / 2!).
To calculate the number of arrangements where the two R's are not together, we subtract the number of arrangements where the two R's are together from the total number of arrangements:
Total arrangements - Arrangements with R's together =
Therefore, there are 37800 ways to arrange the letters of the word "ARRANGE" such that the two R's are never together.
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