Bill and matt are discussing their children. bill tells matt that he has 3 kids, and the product of their ages is 72. he also tells him the sum of the ages of his kids. matt says there is still not enough information to correctly guess the ages of bill's kids. finally, bill says, my youngest child's name is joy. matt can then correctly guess. what are the ages?

Bill tells Matt that he has 3 kids. This means that in this item, we have three unknowns, those are the ages of the three kids. First, the product of the ages is 72.
                            (x)(y)(z) = 72
Also, for the sum of their ages, we have
                              x + y + z =?

There are three variables in the equation which means that 3 equations should also be presented to determine the ages of the three children.