Question is a) State the pigeonhole priciple. b) Explain how the pigeonhole principle can be used to show that among any 11 integers, at least two must have the same digit. Solution a) The pigeonhole principle states that if n pigeons are placed into m pigeonholes, with n>m, then at least one of the pigeonholes must have more than one pigeon. Ex: If there are 366 children in a room, then at least 2 of the kids will share the same birthday (why? because at worst, let\'s assume 365 of the kids all have different birthdays. Then the last kid, the 366th kid, must fall on a day already taken). b) Suppose we have 11 random digits. At worst, we know that the numbers 0, 1, 2, 3, ..., 9 are unique (i.e. don\'t have the same digit). There are ten numbers in that list. No matter what random number is selected for the 11th number, it must contain at least ONE of the digits I just listed. Hence, at least two must have the same digit..