How many ways are there to choose positive integers a, b and c, not necessarily
distinct, so that a + b < c and c ≤ 5?
To find the number of ways to choose positive integers a, b, and c such that a + b < c and c ≤ 5, we can consider the values of c ranging from 1 to 5.
For c = 1, there is no possibility of choosing a and b such that a + b < c, since the sum of two positive integers can never be less than 1. So for c = 1, there are 0 ways to choose a, b, and c.
For c = 2, there is also no possibility of choosing a and b such that a + b < c, since the sum of two positive integers can never be less than 2. So for c = 2, there are 0 ways to choose a, b, and c.
For c = 3, we have the following possibilities:
a = 1, b = 1: 1 + 1 = 2 (not less than 3)
a = 1, b = 2: 1 + 2 = 3 (not less than 3)
a = 2, b = 1: 2 + 1 = 3 (not less than 3)
So there are 0 ways to choose a, b, and c when c = 2.
For c = 4, we have the following possibilities:
a = 1, b = 1: 1 + 1 = 2 (less than 4)
a = 1, b = 2: 1 + 2 = 3 (less than 4)
a = 2, b = 1: 2 + 1 = 3 (less than 4)
a = 1, b = 3: 1 + 3 = 4 (not less than 4)
a = 3, b = 1: 3 + 1 = 4 (not less than 4)
a = 2, b = 2: 2 + 2 = 4 (not less than 4)
a = 1, b = 4: 1 + 4 = 5 (not less than 4)
a = 4, b = 1: 4 + 1 = 5 (not less than 4)
a = 3, b = 2: 3 + 2 = 5 (not less than 4)
a = 2, b = 3: 2 + 3 = 5 (not less than 4)
So there are 3 ways to choose a, b, and c when c = 4.
For c = 5, we have the following possibilities:
a = 1, b = 1: 1 + 1 = 2 (less than 5)
a = 1, b = 2: 1 + 2 = 3 (less than 5)
a = 2, b = 1: 2 + 1 = 3 (less than 5)
a = 1, b = 3: 1 + 3 = 4 (less than 5)
a = 3, b = 1: 3 + 1 = 4 (less than 5)
a = 2, b = 2: 2 + 2 = 4 (less than 5)
a = 1, b = 4: 1 + 4 = 5 (not less than 5)
a = 4, b = 1: 4 + 1 = 5 (not less than 5)
a = 3, b = 2: 3 + 2 = 5 (not less than 5)
a = 2, b = 3: 2 + 3 = 5 (not less than 5)
a = 1, b = 5: 1 + 5 = 6 (not less than 5)
a = 5, b = 1: 5 + 1 = 6 (not less than 5)
a = 4, b = 2: 4 + 2 = 6 (not less than 5)
a = 2, b = 4: 2 + 4 = 6 (not less than 5)
a = 3, b = 3: 3 + 3 = 6 (not less than 5)
So there are 6 ways to choose a, b, and c when c = 5.
In total, there are 0 + 0 + 3 + 6 = 9 ways to choose positive integers a, b, and c such that a + b < c and c ≤ 5.