(a) Treat the two As as a single unit and the two Ls as a single unit. This gives us 7 units to arrange: AA, LL, I, G, A, T, O, R. The number of arrangements is given by:

\(7! = 5040\)

(b) First, calculate the total number of arrangements of the 9 letters:

\(\frac{9!}{2!2!} = 90720\)

Now, calculate the number of favorable arrangements where the two Ls are together and there are exactly 6 letters between the two As. Consider the arrangement of 6 letters including Ls between As:

\(5! \times 5 \times 2 = 1200\)

The probability is then:

\(\frac{1200}{90720} = 0.0132\)