# Five men and 5 women are ranked according to their scores on an examination

S
40 points

Five men and 5 women are ranked according to their scores on an examination. Assume that no two scores are alike and all 10! possible rankings are equally likely. Let X denote the highest rank-ing achieved by a woman. (For instance, X = 1 if the top-ranked person is female.) Find P{X = i}, i = 1, 2, 3, . . . , 8, 9, 10.

Five men
SnakeOil

8.7k points

$$P(X=1)=\frac{C(5,1) \cdot C(9,4) \cdot 4! \cdot 5!}{10!}=\frac{1}{2}$$

the term $C(5,1)$ is the number of ways to choose the top ranked female, then $C(9,4)$ ways to choose the four remaining women and 4! ways to arrange them, and 5! ways to arrange the men.

Next we have P(X=2),

$$P(X=2)=\frac{C(5,1) \cdot C(8,4) \cdot 4! \cdot C(5,4) \cdot 4!}{10!}=\frac{5}{18}$$

we have $C(5,1)$ ways to pick the woman at position 2. Then we have $C(8,4)$ to pick the remaining slots to the right of this woman and 4! ways to arrange them. There are then 8-4=4 slots left to place the men into and $C(5,4)$ ways to pick the four men to fill these spots and 4! ways to order them.

$$P(X=3)\frac{C(5,1)\cdot C(7,4) \cdot 4! \cdot C(5,3) \cdot 3! \cdot 2!}{10!}=\frac{5}{36}$$

proceed in this manner making sure that your probabilities add up to 1 in the end.

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