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June 2021 p52 q6
2684
(a) Find the total number of different arrangements of the 8 letters in the word TOMORROW.
(b) Find the total number of different arrangements of the 8 letters in the word TOMORROW that have an R at the beginning and an R at the end, and in which the three Os are not all together.
Solution
(a) The word TOMORROW consists of 8 letters where T, M, and W are unique, O appears 3 times, and R appears 2 times. The total number of different arrangements is given by:
\(\frac{8!}{2!3!} = \frac{40320}{12} = 3360\)
(b) To find the number of arrangements with R at the beginning and end, and the Os not all together, we first consider the arrangement of the remaining 6 letters (T, O, M, O, W, O). The total arrangements of these 6 letters is:
\(\frac{6!}{3!} = 120\)
Next, we subtract the cases where all Os are together. Treating the three Os as a single unit, we arrange T, M, W, and the O unit:
\(4! = 24\)
Thus, the number of arrangements where the Os are not all together is:
