Final answer:
To find the number of permutations of all the letters in a name, you multiply every whole number less than or equal to the number of letters down to 1 (factorial). For 7 letters, this is 7!, or 5040 permutations. For 8 letters, it's 8!, or 40320 permutations.
Step-by-step explanation:
The question is asking for the number of permutations of all the letters in each name. In mathematics, a permutation is an arrangement of items in a particular order. The number of permutations can be found using the factorial function, denoted by an exclamation point. For example, let's say a name has n distinct letters, then the number of different ways to arrange those letters would be n! (n factorial), which is n multiplied by every whole number less than it down to 1.
If a name contains 7 letters, then the number of permutations is 7!, which equals 5040. If a name has 8 letters, the permutations would be 8!, equating to 40320. Therefore, by calculating the factorials for names with 7 and 8 letters, we can determine the correct answer. Following this logic, if both names have 7 letters, the permutations for each would be 5040, and if one name has 7 letters and the other 8, the permutations would be 5040 and 40320, respectively.