2015
Problem - 2028
A hand of four cards of the form $(c, c, c + 1, c + 1)$ is called a $tractor$. Vinjai has a deck consisting of four of each of the numbers $7$, $8$, $9$ and $10$. If Vinjai shuffles and draws four cards from his deck, compute the probability that they form a tractor.
Consider all the cards with distinct (i.e. imagine they are spade, heart, club, and diamond) and all the drawn cards are ordered. Then there are $C_{16}^4$ to draw $4$ cards. Among them, there are three possible combinations of tractor each of which has $C_4^2\times C_4^2$ ways to obtain. Hence, the result is $$\frac{3\times C_4^2\times C_4^2}{C_{16}^4}=\boxed{\frac{27}{455}}$$