Question

You have five rods, A, B, C, D and E, with lengths of 10

cm, 20
cm, 30
cm, 40
cm and 50
cm. A

B

C

D

E

You pick up three rods at random and, if it is possible, make them into a triangle. For example if you pick B, C and D you can make make a triangle. If you pick A, B and E you cannot make a triangle. There are 10
different combinations of three rods that can be made from the five rods, for example, ABC, ABE, DCE. A Write down the 10
different combinations. B Is it more likely that you will pick three that will make a triangle, or you will pick three rods that will not make a triangle? Explain your answer fully

Answer 1

• ABC, ABD, ABE, ACD, ACE, ADE, BCD, BCE, BDE, CDE
• most likely: won't make a triangle

Explanation:

Given five rods {A, B, C, D, E} of lengths {10, 20, 30, 40, 50} cm, you want a list of possible combinations of 3 rods and whether it is more likely than not that they will form a triangle.

A. Combinations

The possible combinations of rods is the number of subsets of 5 items taken 3 at a time. That number is (5·4·3)/(3·2·1) = 10. The possible combinations are ...

ABC, ABD, ABE, ACD, ACE, ADE, BCD, BCE, BDE, CDE

B. Triangle

Of these combinations, only three (3) of them, BCD, BDE, and CDE, will form a triangle. The corresponding side lengths are {20, 30, 40}, {20, 40, 50}, {30, 40, 50} in centimeters.

If 3 of 10 combinations form a triangle, then 7/10 will not. If you choose rods at random, it is more like they will not form a triangle.

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Additional comment

The triangle inequality requires the sum of the two shortest lengths exceed the longest length.

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