Question

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?

Answer 1

(a) Take the digits 9876543210 and remove any 3 of them, that gives you 120 possible numbers

(b) To get a number like this you have to remove the 0, do not remove the 1, and remove 2 other digits, that gives you 28 different numbers.
28/120 = 23.333%

(c) the possible values times the number of ways you can get them:
6*20 => 120 5*40
=> 200 4*40
=> 160 3*20
=> 60 total
=> 540