# How to solve this probability problem related to numbers of a lottery?

In a simple lottery the organiser chooses three numbers at random without replacement from the numbers 1 to 5. Players also choose three numbers without replacement from the numbers 1 to 5.

Calculate the probability that a player matches two or three of the organiser’s numbers.

What I tried-

5C3 ways to choose three numbers i.e 10 ways to choose three numbers. After that I can't proceed.

• As there are only ten combinations for each, you could make a 10×10 table and count the hits by hand; in the process you might get an insight. – Anton Sherwood Nov 10 '15 at 9:42

• The probability that a player matches exactly $\color\red2$ numbers, which is $\dfrac{\binom{3}{\color\red2}\cdot\binom{5-3}{3-\color\red2}}{\binom{5}{3}}=\dfrac{3}{5}$
• The probability that a player matches exactly $\color\red3$ numbers, which is $\dfrac{\binom{3}{\color\red3}\cdot\binom{5-3}{3-\color\red3}}{\binom{5}{3}}=\dfrac{1}{10}$