Answer:
1/82.
Explanation:
In a cross country race with 40 athletes, 10 of them are on the same team. The probability that the top four finishers are all from that same team is given by the expression 10P4/40P4.
The expression 10P4 represents the number of ways that the top four finishers can be chosen from the team of 10 athletes. Since the order in which the athletes finish does not matter, this is equivalent to choosing 4 athletes from a group of 10 without regard for order, which can be done in 10!/(4!6!) = 210 ways.
The expression 40P4 represents the number of ways that the top four finishers can be chosen from the entire group of 40 athletes. Since the order in which the athletes finish does not matter, this is equivalent to choosing 4 athletes from a group of 40 without regard for order, which can be done in 40!/(4!36!) = 34,650 ways.
Therefore, the probability that the top four finishers are all from the same team is given by the expression 10P4/40P4 = 210/34,650 = 1/164. This can be simplified to 1/82 by dividing the numerator and denominator by 2, so the final answer is 1/82.