Pat writes all the 7-digit numbers in which all the digits are different and each digit is greater than the one to its right (so the tens digit is greater than the units, the hundreds greater than the tens,
and so on). For example, 9,865,320 is one of the numbers that Pat writes down.
(a) How many numbers does Pat write down?
(b) One of Pat's numbers is chosen at random. What is the probability that the tens digit is a 1?
(c) One of Pat's numbers is chosen at random. What is the probability that the middle (thousands) digit is a 5?
Note: of course, you could solve this problem by repeating Pat's experiment and writing down all of the numbers. But don't do that -- figure out the answers without needing to write down all of the numbers!