2nd Mexico 1988 Problem A4 How many ways can we find 8 integers a 1 , a 2 , ... , a 8 such that 1 ≤ a 1 ≤ a 2 ≤ ... ≤ a 8 ≤ 8?
15C7 = 15!/(8!7!) = 6435 Solution Arrange 8 black balls and 7 white balls in a row. Then the 7 white balls divide the line into 8 parts corresponding to a i = 1, 2, ... , 8. The number of black balls in each part gives the number of ai with that value. For example WBBWWBWBWBBWWBB corresponds to a 1 = a 2 = 2, a 3 = 4, a 4 = 5, a 5 = a 6 = 6, a 7 = a 8 = 8. 2nd Mexican 1988 © John Scholes jscholes@kalva.demon.co.uk 21 February 2004 Last corrected/updated 21 Feb 04