All the sides of a triangle are integers, and the perimeter is 15. How many different possible triangles are there? (Assume that the triangle is non-degenerate. Two triangles are considered the same if they are congruent.)
+592 we can say that one side length is a, another b, and the last c. We know that:
a+b>c
a+c>b
c+b>a
a+b+c=15
We also know that order matters. if a=1, b and c must be 7. if a=2, b must be 7, and c must be 6. we can continue counting:
a=1,b=7,c=7
a=2,b=7,c=6
a=3,b=6,c=6
a=3,b=7,c=5
a=4,b=6,c=5
a=4,b=7,c=4
a=5,b=5,c=5
those are all the possible solutions, so $\boxed{7}$ is our answer
+592 we can say that one side length is a, another b, and the last c. We know that:
a+b>c
a+c>b
c+b>a
a+b+c=15
We also know that order matters. if a=1, b and c must be 7. if a=2, b must be 7, and c must be 6. we can continue counting:
a=1,b=7,c=7
a=2,b=7,c=6
a=3,b=6,c=6
a=3,b=7,c=5
a=4,b=6,c=5
a=4,b=7,c=4
a=5,b=5,c=5
those are all the possible solutions, so $\boxed{7}$ is our answer
