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June 2022 p51 q1
2737
(a) Find the number of different arrangements of the 8 letters in the word DECEIVED in which all three Es are together and the two Ds are together.
(b) Find the number of different arrangements of the 8 letters in the word DECEIVED in which the three Es are not all together.
Solution
(a) Consider the three Es as a single unit and the two Ds as another single unit. This reduces the problem to arranging the units: {EEE, DD, C, I, V}. There are 5 units to arrange, so the number of arrangements is given by:
\(5! = 120\)
(b) First, find the total number of arrangements of the letters in DECEIVED. The word has 8 letters with repetitions of E and D. The total number of arrangements is:
\(\frac{8!}{2!3!} = 3360\)
Next, find the number of arrangements where all three Es are together. Treat the three Es as a single unit, reducing the problem to arranging {EEE, D, D, C, I, V}. The number of arrangements is:
\(\frac{6!}{2!} = 360\)
Therefore, the number of arrangements where the three Es are not all together is:
