Question number two, how many triangles are there with a perimeter of seven and all side lengths expressed by integers? So if we draw an example triangle, we have here three sides that we can label with letters A, B, and C, and what we need is that the perimeter, so the sum of these lengths here, is equal to seven, and we need A, B, and C to be integers, so positive numbers. And the first thing we should do is write seven as a sum of three positive numbers, so there are several ways to do that. We can have one plus one plus five, seven can also be written as one plus two plus four, we can have a one plus three plus three, and finally two plus three plus two. And so here we notice that order does not matter. We can rotate or flip around our triangle so the order of the sides does not really matter as long as we have three positive integers that add up to seven, however what does matter is the triangle inequality has to be respected, but the sum of any two sides must at least exceed the third. So if we check our possibilities here, we have that the first partition of seven into three integers will not work, two plus five is seven, but if we take the two sides with length one, their sum is less than five, likewise the second possibility will not work, one plus two is three, which is less than four, and looking at the third option here, one plus three is four, which is greater than the remaining side, so this one works, and then here if we take two plus two or two plus three, the sum is also greater than the remaining side, so out of the three ways we can partition seven into three positive integers, only two of those possibilities actually give us a valid triangle, so the answer here is two, there are two such triangles.

integer sides

triangle inequality

perimeter seven

valid triangles

combinations