Jerry is creating a 9-character password for his tablet using symbols from the following categories: capital letters, lowercase letters, numbers, punctuation, and arithmetic operations. The password must include at least one symbol from each category and cannot have consecutive symbols from the same category. If the 1st and 5th positions are both question marks, lowercase letters must directly follow every punctuation mark, and numbers are used more frequently than any other type of symbol, then which of the following characters could be used for the 7th and 9th positions? Select a possible character for the 7th and 9th positions of the password. Make only two selections, one in each column.

Official Solution: Given that the 1st and 5th positions are both question marks, and lowercase letters must directly follow every punctuation character, we would have the following case (using lowercase a as an example for a lowercase character): 1 2 3 4 5 6 7 8 9 ? a ? a There are 5 more positions to fill, and we must still use characters from 3 categories: capital letters, numbers, and arithmetic operations. Since numbers are used more frequently than any other type of character, there must be three numbers, one capital letter, and one arithmetic operation. Additionally, as consecutive characters cannot be from the same category, a number must occupy either the 3rd or 4th position, as well as the 7th and 9th positions. Therefore, both the 7th and 9th positions must be numbers. For example, we can have the following case: 1 2 3 4 5 6 7 8 9 ? a 1 + ? a 9 A 9 Correct answer: 7th character "9" 9th character "9" GMAT-Club-Forum-m74rauz0.png