Question

Six runners compete in a track race. Three are from Red Hook and three are from Rhinebeck.
(a) In how many different orders can these runners finish if the first three are from Red Hook and the last three are from Rhinebeck?
(b) What is the probability, if all of the runners are equally talented, that the first three finishers would be from Red Hook by chance alone?
(c) What is the probability, if all of the runners are equally talented, that the first three finishers would be from the same school?

Answer 1

(a) The number of permutations of 3 taken 3 at a time is 3P3. Therefore the number of different orders with 1st three from Red Hook and the last three from Rhinebeck is:
3P3 + 3P3
(b)\[P(1st* three* ex *RH)=\frac{\left(\begin{matrix}3 \\ 3\end{matrix}\right)\left(\begin{matrix}3 \\ 0\end{matrix}\right)}{\left(\begin{matrix}6 \\ 3\end{matrix}\right)}=\frac{3\times 3\times 2}{6\times 5\times 4}\]
(c) Same as (b)