(a) The word REFRIGERATOR has 12 letters with repetitions: R appears 3 times, and E appears 2 times. The number of arrangements is given by:

\(\frac{12!}{3! \times 2!}\)

Calculating this gives:

\(\frac{479001600}{6 \times 2} = 9979200\)

(b) If the Rs must all be together, treat the 3 Rs as a single unit. This gives us 10 units to arrange (9 other letters + 1 unit of Rs). The number of arrangements is:

\(\frac{10!}{2!}\)

Calculating this gives:

\(\frac{3628800}{2} = 181440\)