Question

A bag contains six sticks of the following lengths - 1 cm, 3 cm, 5 cm, 7 cm, 11 cm and 13 cm. three sticks are drawn at random from the bag. what is the probability that we can form a triangle with those sticks?

Answer 1

1 if the sticks can overlap

Answer 2

Answer:

Probability that we can form a triangle with those sticks is $\frac{1}{4}$

Step-by-step explanation:

Given : A bag contains six sticks of the following lengths - 1 cm, 3 cm, 5 cm, 7 cm, 11 cm and 13 cm. Three sticks are drawn at random from the bag.

To find : What is the probability that we can form a triangle with those sticks?

Solution :

We know, the property of triangle that every pair of the three straws drawn  must have lengths whose sum is greater than the third side.

A bag contains six sticks of the following lengths - 1 cm, 3 cm, 5 cm, 7 cm, 11 cm and 13 cm.

i.e if we choose the longest two straws, 13cm and 11cm,

the third straw can be 3cm, 5cm or 7cm.  

Similarly, there are only five successful choices of three so that the  sum of every pair of straw's lengths is greater than the third  straw's length.

They are  

1. 13,11,7 cm

2. 13,11,5 cm

3. 13,11,3 cm

4. 11,7,5 cm

5. 7,5,3 cm

Favorable outcomes = 5

So, There are $^6C_3$ ways to draw three straws.

i.e.  $^6C_3=\frac{6!}{3!(6-3)!}$

$^6C_3=\frac{6\times 5\times 4\times 3!}{3!\times 3\times 2\times 1}$

$^6C_3=20$

Total number of outcome = 20

Probability that we can form a triangle with those sticks is

$P=\frac{5}{20}$

$P=\frac{1}{4}$

Therefore, Probability that we can form a triangle with those sticks is $\frac{1}{4}$