There are 20 empty present boxes numbered from 1 to 20 are placed on a shelf, there are 4 men standing in front of the shelf. Each one asked to pick in his mind 3 numbers without telling any one then write down his selection. How can I know the following :
- Probability of choosing a mutual number by any two or more men.
- After finishing the selection, what is the probability for a different man, not one of the 4 men, to select the same boxes selected by them.
The probability that some two men choose the same number is the complement of the probability that no two men choose the same number.
The number of ways to choose the three men's numbers such that they are all different is
$$\binom{20}{3}\binom{17}{3}\binom{14}{3}\binom{11}{3} = 1140\cdot 680\cdot 364\cdot 165$$
The number of ways to choose the same numbers with no restriction is
$$\binom{20}{3}\binom{20}{3}\binom{20}{3}\binom{20}{3}= 1140\cdot 1140\cdot 1140\cdot 1140$$
The probability of the numbers being different is therefore the first expression above, divided by the second expression:
$$P =\frac{1140\cdot 680 \cdot 364 \cdot 165}{1140^4}=\boxed{\frac{17017}{617310 }}$$