(i) The word CAMERAMAN consists of 9 letters where A appears 3 times and M appears 2 times. The number of arrangements without restrictions is given by:

\(\frac{9!}{3! \times 2!} = \frac{362880}{6 \times 2} = 30240\)

(ii) If the As occupy the 1st, 5th, and 9th positions, we arrange the remaining 6 letters (C, M, E, R, M, N). The number of arrangements is:

\(\frac{6!}{2!} = \frac{720}{2} = 360\)

(iii) If there is exactly one letter between the Ms, consider the block M_X_M where X is any letter. We have 7 positions for this block among the remaining letters (A, A, A, C, E, R, N). The number of arrangements is:

Method 1: Arrange the 7 letters (excluding M) and multiply by 7 positions for the block:

\(\frac{7!}{3!} \times 7 = \frac{5040}{6} \times 7 = 5880\)

Method 2: Choose a letter between Ms and arrange:

\(\frac{6!}{2!} \times 7 + \frac{6!}{3!} \times 7 = 2520 + 3360 = 5880\)