We need to select 6 letters from the word CATERPILLAR, which contains the letters C, A, T, E, R, P, I, L, L, A, R. The selection must include both Rs, at least one A, and at least one L.

Method 1:

1. Fix R, R, A, L. We need to choose 2 more letters from the remaining 5 letters: C, T, E, P, I.
2. Calculate the combinations: \\(^5C_2 = 10\).
3. Fix R, R, A, L, L. Choose 1 more letter from C, T, E, P, I: \\(^5C_1 = 5\).
4. Fix R, R, A, A, L. Choose 1 more letter from C, T, E, P, I: \\(^5C_1 = 5\).
5. Fix R, R, A, A, L, L. No more letters needed: 1 way.

\(Total = 10 + 5 + 5 + 1 = 21.\)

Method 2:

1. Fix R, R, A, L. Choose 2 more letters from the remaining 7 letters: C, T, E, P, I, A, L.
2. Calculate the combinations: \\(^7C_2 = 21\).

Both methods give the total number of selections as 21.