(a) To arrange the letters of SUMMERTIME with an E at the beginning and an E at the end, we fix the Es in these positions. This leaves us with 8 letters: S, U, M, M, R, T, I, M. The number of ways to arrange these 8 letters is given by:
\(\frac{8!}{3!}\)
where 3! accounts for the repetition of the letter M. Calculating this gives:
\(\frac{8!}{3!} = \frac{40320}{6} = 6720\)
(b) To find the number of ways to arrange the letters of SUMMERTIME such that the Es are not together, we first find the total number of arrangements of the letters and then subtract the number of arrangements where the Es are together.
The total number of arrangements of the letters is:
\(\frac{10!}{2!3!}\)
where 2! accounts for the repetition of E and 3! for M. Calculating this gives:
\(\frac{10!}{2!3!} = \frac{3628800}{12} = 302400\)
Next, consider the Es as a single unit. This gives us 9 units to arrange: (EE), S, U, M, M, R, T, I, M. The number of ways to arrange these is:
\(\frac{9!}{3!}\)
Calculating this gives:
\(\frac{9!}{3!} = \frac{362880}{6} = 60480\)
Therefore, the number of arrangements where the Es are not together is:
\(302400 - 60480 = 241920\)
