Question 6
Alam has nine cards numbered 1 through 9.
He first takes one odd and one even card, then another odd and even card from those remaining, and arranges the four cards in a row. Rule 1: “The product of each and every pair of odd and even cards selected is always a one-digit number.” Rule 2: the positive difference between the two odd cards must differ from the positive difference between the two even cards. (a) How many four-digit numbers satisfy these rules? (b) How many if the product in Rule 1 is instead a two-digit number less than 20? (c) How many if it is instead a multiple of 3 less than 40?
He first takes one odd and one even card, then another odd and even card from those remaining, and arranges the four cards in a row. Rule 1: “The product of each and every pair of odd and even cards selected is always a one-digit number.” Rule 2: the positive difference between the two odd cards must differ from the positive difference between the two even cards. (a) How many four-digit numbers satisfy these rules? (b) How many if the product in Rule 1 is instead a two-digit number less than 20? (c) How many if it is instead a multiple of 3 less than 40?