How many solutions does a triangle with side lengths a = 4, a = 68 , and b = 10 have?

The easiest way to find if a triangle can be formed is by applying the law of sinus: a/sin(A) = b/sin(B). The range of sin must be [-1, 1].

In our problem:
sin68/4= sin(B)/10
sin(B) = 10/4*sin68 = 2.5sin(68)

Function sin is an increasing function between 0-90 degrees so we can say that 30<68<90 so do sin(30)<sin(68)<sin(90). 

1/2 < sin(68) < 1 . Multiplying by 2.5 we have:
1.25<2.5sin(68)< 2.5
1.25<sin(B)<2.5

But as I said the range of sin must be [-1, 1]. So there is no solution in order to form a triangle(No triangle can be formed).