How many two digit whole numbers can be written as the sum o

How many two digit whole numbers can be written as the sum of two different single digit whole numbers?

Answer: 36 Please show the detail work and explain the problems.

Solution

A positive 2 digit numbers can be written as the sum of two different single digit positive numbers to be selected from 1,2,3,4,5,6,7,8,9 only( as 0, when added to any other single digit number, will yield a single digit numberonly). Now, 2 digit numbers can be selected from 9 in ₉C₂ = 9!/[2! (9-2)!] = 9!/[2*(7!)] = 36 ways. Further, any 2 of the single digit positive numbers 1,2,3,4,5,6,7,8,9, when added to any other single digit positive number from this list, will result in a two digit positive whole number. The same logic applies to negative numbers. Hence, the answer is 36.