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

`\frac{\binom64\binom{42}2}{\binom{48}6}=\boxed{(4,305)/(4,090,504)}`

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

`\dbinom nk=(n!)/(k!(n-k)!)`