Final answer:
To calculate the number of words that can be formed with restrictions on where A's, B's, and C's can appear, we consider the placement of B's in the first group and multiply the permutations for each subsequent group, summing across all possible distributions of Bs in the first group.
Step-by-step explanation:
The question involves finding the number of ways to arrange 6 A's, 6 B's, and 6 C's following specific restrictions. The first 6 letters cannot be A's, the second 6 letters cannot be B's and the third 6 letters cannot be C's. To solve this problem, we can use combinatorial methods and the hint to group different ways according to the number of B's in the first group.
For example, if there are 0 B's in the first group, then we have 6 C's in the first group and therefore must have 6 B's in the second group and 6 A's in the last. This is just one way to arrange them.
If there is 1 B in the first group, then we have 5 C's alongside it, which means there are 5 B's left for the second group. We can select the placement for that single B amongst the 6 positions in the first group in 6 ways. Then, the second group will also contain 5 B's and 1 A, and we can select the position for the A in 6 ways. So for 1 B in the first group, there are 6 × 6 ways.
Continuing this approach, we would calculate the number of arrangements for each possible number of B's in the first group (0 through 5), and sum them to get the total number of arrangements.