The digits 1, 3 and 5 are each used once to form each of the possible three-digit positive integers. The three-digit integers are listed from greatest to least. Which integer is listed fifth?
I need to find the possible three-digit integers that can be formed using the digits 1, 3 and 5, without repeating any of them. I can start by finding the greatest such integer, which is 531, since I want to use the largest digit in the hundreds place, the next largest in the tens place, and the smallest in the units place. To find the next greatest integer, I can keep the hundreds digit as 5, but switch the tens and units digits, so I get 513. To find the next one, I can change the hundreds digit to 3, and use the largest remaining digit, 5, in the tens place, and the smallest remaining digit, 1, in the units place, so I get 351. To find the next one, I can keep the hundreds digit as 3, but switch the tens and units digits, so I get 315. To find the fifth greatest integer, I can change the hundreds digit to 1, and use the largest remaining digit, 5, in the tens place, and the next largest remaining digit, 3, in the units place, so I get 153. Therefore, the integer that is listed fifth is 153. 