SOLUTION: One sports federation runs a council of 14 women and 10 men. The Federation decided to select a small committee from The Board consists of 4 members at random, and elects a chairm

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Question 1207850: One sports federation runs a council of 14 women and 10 men. The Federation decided to select a small committee from
The Board consists 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?
Answer by ikleyn(52794) About Me  (Show Source):
You can put this solution on YOUR website!
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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 highlight%28cross%28consists%29%29 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?
~~~~~~~~~~~~~~~~~~~~ 

The total number of members of the council is 14+10 = 24.


The number of all possible quadruples to form from 24 persons is  

    C%5B24%5D%5E4 = %2824%2A23%2A22%2A21%29%2F%281%2A2%2A3%2A4%29 = 10626.


    +-----------------------------------------------------+
    |   Now let's calculate the number of all quadruples  |
    |   that are compounded as described in the problem.  |
    +-----------------------------------------------------+


The number of ways to select 3 women from 14 women is  C%5B14%5D%5E3 = 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 = 3640%2F10626 = 0.3426  (rounded).

Solved.