(a) The word CATERPILLAR consists of 11 letters where C, A, T, E, P, I, L, L, A, R, R are present. The letters A, L, and R are repeated. The formula for permutations of letters with repetitions is given by:

\(\frac{11!}{2!2!2!}\)

Calculating this gives:

\(\frac{11!}{2!2!2!} = \frac{39916800}{8} = 4989600\)

(b) To find the number of arrangements with R at the beginning and end, and the two As not together, we use two methods:

Method 1:

Arrange the 7 letters CTEPILL (excluding the two Rs and two As):

\(\frac{7!}{2!}\)

Calculate the number of ways to place As in non-adjacent places:

\(\binom{8}{2}\)

The total number of arrangements is:

\(\frac{7!}{2!} \times \binom{8}{2} = 2520 \times 28 = 70560\)

Method 2:

Total arrangements with R at the beginning and end:

\(\frac{9!}{2!2!}\)

Arrangements with R at ends and As together:

\(\frac{8!}{2!}\)

With As not together:

\(\frac{9!}{2!2!} - \frac{8!}{2!} = 90720 - 20160 = 70560\)