mathschallenge.net

Problem

For each of the numbers: 41, 83, 32, the first digit is greater in value than the second digit.

How many 2-digit numbers have this property?

Solution

If we begin to list the numbers in groups: 10; 20,21; 30,31,32; 40,41,42,43; ... ; 90,91,92,93,94,95,96,97,98 ; we can see that the total number of 2-digit numbers, for which the first digit is greater than the second digit, will be 1 + 2 + ... + 9 = 45.

How many 3-digit numbers exist for which the first digit is greater in value than both the second digit and the third digit?
Can you generalise for n-digit numbers?
What about 3-digit numbers for which the first digit is greater than the sum of the second and third digits?