(a) The word REQUIREMENT has 11 letters with the letters R and E repeating. The total number of arrangements is given by:

\(\frac{11!}{2!3!}\)

Calculating this gives:

\(\frac{39916800}{12} = 3326400\)

(b) Treat the two Rs as a single unit and the three Es as another single unit. This gives us 9 units to arrange (RR, EEE, Q, U, I, M, N, T). The number of arrangements is:

\(9!\)

Calculating this gives:

\(362880 = 40320\)

(c) To have exactly three letters between the two Rs, consider the Rs as a block with three spaces between them. The remaining 8 letters can be arranged in the remaining positions. The number of ways to choose 3 letters to place between the Rs is:

\(\binom{8}{3}\)

Then, arrange the remaining 8 letters:

\(8!\)

Finally, multiply by the number of ways to arrange the 3 letters between the Rs:

\(3!\)

The total number of arrangements is:

\(\binom{8}{3} \times 8! \times 3! = 56 \times 40320 \times 6 = 423360\)