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One sports federation runs a council of 14 women and 10 men. The Federation decided
to select a small committee from the board consisting of 4 members at random,
and elects a chairman, a secretary, and two treasurers. What is the probability
that the committee consists of 3 women, one of whom is the chairperson of the committee,
and one man is the secretary of the committee?
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The total number of members of the council is 14+10 = 24.
The number of all possible quadruples to form from 24 persons is
= = 10626.
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| Now let's calculate the number of all quadruples |
| that are compounded as described in the problem. |
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The number of ways to select 3 women from 14 women is = 364.
How these three women will occupy/distribute the three positions inside the committee -
this fact is excessive for this problem and does not make influence on the number of women' triples.
The number of ways to select 1 man from 10 men is 10, obviously.
This man will inevitably occupy the position of the secretary -
so, this info is excessive and does not make influence on further solution.
Thus, the number of all different committees as described in the problem is 364*10 = 3640.
Now the probability under the problem's question is
P = = 0.3426 (rounded).
Solved.