Answer each question given that Victor has five straws with lengths of 3 cm, 4 cm, 5 cm, 6 cm, and 12 cm
1. How many different triangles can Victor make with the straws
2. How many different triangles can Victor make that have a perimeter that is divisible by 3
3. There are 10 different combinations of straws. What is the probability that if Victor chooses three straws at random he will be able to make a triangle (Hint: number of choices that will make a triangle divided by total number of ways to choose 3 straws from a group of 5)
Total Number of Triangles:
In a triangle, the length any one sides must be less than the sum of lengths of the other two sides.
We find that this condition cannot be fulfilled at all when straw with length 12 cm is taken as one of the side.
Also, this condition is met when any three of the other 4 straws are selected.
To make different triangles out of these four straws measuring 3 cm, 4 cm, 5 cm, and 6 cm, we need to select three straws. Or, put in another way we have to remove 1 straw from a total of 4. This can be done in 4 different ways.
Therefore, Victor can make 4 different triangles from the given straws.
Triangles with Perimeter Divisible by 3:
Perimeter of a triangle made out of any 3 of the 4 straws will be divisible by 3 only when both the straws of length 4 cm and 4 cm are used for making the triangle. Please note that it is not possible to make a triangle without both of these, and if only one of these is selected in combination of the other two with lengths of 3 cm and 6 cm, the perimeter will not be divisible by 3.
Thus a triangle with perimeter divisible by 3 can be made by removing either of the straw with length of 3 cm or 6 cm. This can be done in 3 ways.
Therefore, Victor can make 2 triangles wit perimeter divisible by 3.
Probability of Choosing 3 Straws Making a Triangle:
Victor will be able to make a triangle when either of the two straws left out is 12 cm straw.
The probability of this is given by: 1- (0.2 + 0.8*0.25) = 0.6
Thus probability is 0.6 that victor will be able to make a triangle.
1)(3,4,5),(3,4,6), (3,5,6) (4,5,6) are the possible triangles, as in a triangle sum of any two sides is always grater than the remaing sides . If we goup in any other way, this is not possible.
2)(3,4,5) and (4,5,6) are the 2 triangles, whose perimetrs are 12 and 15 are divisible by 3.
3)Number of possiblities of making triangles as at (1)/5C3 = 4/10 =0.4 is the required probability.