Triangles?

A bag contains some sticks with lengths three feet, four feet, five feet, six feet, and seven feet.  Three sticks are chosen at random.

a) What is the probability that a triangle can be formed with the three chosen sticks?

 

3,4,5,6,7   there are 5C3 ways to chose three different ones  5C3=10  that is not many

If 3 are the same then there are 5*4C2 ways = 30 ways      

If they are all the same then there are 5 ways 

There are 45 ways to chose the 3 sticks

 

Also the sticks will from a triangle is the sum of the 2 little ones is longer than the big one. 

So

3,3,6

3,3,7

3,4,7 

are the only three selections that will not form a triangle

 

So

P(a triangle can be formed)= 42/45

 

b) What is the probability a right triangle can be formed with the three sticks?

 

The three sticks will from a right angle if the lengths of the sides are a pythagorean triad.  3,4,5 is the only one. 

 

so

P(forming a right angled triangle) = 1/45

 

P(forming a right angled triangle given that a triangle is able to be formed) = 1/42