Final answer:
The number of permutations that satisfy the given conditions is 18.
Step-by-step explanation:
To find the number of permutations that satisfy the given conditions, we need to consider the positions of each letter and their respective restrictions. The first appearance of 'a' must come before the first appearance of 'b', which must come before the first appearance of 'c', and so on. We can divide the problem into three groups: the 'a', 'b', and 'c' group; the 'd', 'e', and 'f' group; and the 'g' group (if applicable).
The 'a', 'b', and 'c' group has three options for arrangement: 'abc', 'acb', 'bac'.
The 'd', 'e', and 'f' group has six options for arrangement: 'def', 'dfe', 'edf', 'efd', 'fde', 'fed'.
The 'g' group (if applicable) has one option for arrangement: 'g'.
Therefore, the total number of permutations that satisfy the given conditions is 3 x 6 x 1 = 18.