Question

In a lottery game, a player picks six numbers from 1 to 48. If 4 of those 6 numbers match those drawn, the player wins third prize. What is the probability of winning this prize? (Give your answer as a fraction.)

Answer 1

There are $\binom{48}6=12,271,512$ ways of drawing 6 numbers between 1 and 48.

Of the 6 drawn numbers, there are $\binom64=15$ ways of drawing 4 matching numbers, and $\binom{42}2=861$ ways of drawing any 2 non-matching numbers.

Hence the probability of winning the prize is

$\dfrac{\binom64\binom{42}2}{\binom{48}6}=\boxed{\dfrac{4,305}{4,090,504}}$

Note: In case you're unfamiliar with the notation,

$\dbinom nk=\dfrac{n!}{k!(n-k)!}$