Joanna makes as many triangles as she can using combinations of three of these straws. Determine how many different triangles she can make. What combinations of straws does she use? Show your work.
To determine the number of different triangles Joanna can make, we need to consider the number of unique combinations of three straws. Let's assume that Joanna has n straws.
The number of combinations of three straws out of n is denoted by nC3 and can be calculated using the formula:
nC3 = n! / (3! * (n-3)!)
where n! represents the factorial of n.
Now let's consider different scenarios to determine how many triangles Joanna can make:
1) If Joanna has less than three straws, she cannot make any triangles since a triangle requires at least three sides.
2) If Joanna has exactly three straws, then she can make only one triangle since all three straws will form a closed shape without intersecting each other.
3) If Joanna has more than three straws, we can use the formula mentioned earlier to calculate the number of combinations.
For example, if Joanna has five straws (n = 5):
nC3 = 5! / (3! * (5-3)!)
= 5! / (3! * 2!)
= (5 * 4 * 3 * 2 * 1) / ((3 * 2 * 1) * (2 * 1))
= (120) / (6 * 2)
= 10
So, Joanna can make 10 different triangles using five straws, and the combinations of straws she can use are:
- (1, 2, 3)
- (1, 2, 4)
- (1, 2, 5)
- (1, 3, 4)
- (1, 3, 5)
- (1, 4, 5)
- (2, 3, 4)
- (2, 3, 5)
- (2, 4, 5)
- (3, 4, 5)
By applying the same method to other numbers of straws, you can determine the number of different triangles she can make for each scenario.
To determine how many different triangles Joanna can make using combinations of three straws, we need to consider the possible combinations of straws.
Let's assume Joanna has n straws.
To form a triangle, the sum of any two sides must be greater than the third side according to the triangle inequality theorem.
Here's how we can find the combinations of straws Joanna can use to form triangles:
1. Start by selecting three straws out of the given n straws. This can be done using the combination formula: C(n, 3) = n! / (3!(n-3)!).
2. Calculate the value of C(n, 3) and determine the number of combinations.
Let's consider an example:
Suppose Joanna has 6 straws.
Using the formula C(6, 3) = 6! / (3!(6-3)!) = 20, we find that there are 20 different combinations of straws she can use to form triangles:
1. (1, 2, 3)
2. (1, 2, 4)
3. (1, 2, 5)
4. (1, 2, 6)
5. (1, 3, 4)
6. (1, 3, 5)
7. (1, 3, 6)
8. (1, 4, 5)
9. (1, 4, 6)
10. (1, 5, 6)
11. (2, 3, 4)
12. (2, 3, 5)
13. (2, 3, 6)
14. (2, 4, 5)
15. (2, 4, 6)
16. (2, 5, 6)
17. (3, 4, 5)
18. (3, 4, 6)
19. (3, 5, 6)
20. (4, 5, 6)
So, Joanna can make 20 different triangles using the combinations of three straws from a set of 6 straws.