The number of ways to form a committee consisting of men and women depends on the total number of men and women available and the number of men and women required in the committee. The general approach uses combinations:
- To choose rrr men from mmm men, use the combination (mr)\binom{m}{r}(rm).
- To choose sss women from www women, use the combination (ws)\binom{w}{s}(sw).
- The total number of ways to form the committee is the product (mr)×(ws)\binom{m}{r}\times \binom{w}{s}(rm)×(sw).
Example 1:
Forming a committee of 3 men and 2 women from 7 men and 5 women:
- Number of ways to choose men: (73)=35\binom{7}{3}=35(37)=35
- Number of ways to choose women: (52)=10\binom{5}{2}=10(25)=10
- Total ways: 35×10=35035\times 10=35035×10=350
Example 2:
Forming a committee of 7 men and 4 women from 9 men and 6 women:
- Number of ways to choose men: (97)=(92)=36\binom{9}{7}=\binom{9}{2}=36(79)=(29)=36
- Number of ways to choose women: (64)=(62)=15\binom{6}{4}=\binom{6}{2}=15(46)=(26)=15
- Total ways: 36×15=54036\times 15=54036×15=540
Additional Notes:
- If there are restrictions (e.g., certain members cannot serve together), the invalid combinations must be subtracted from the total.
- If the committee requires at least one woman or other conditions, those must be accounted for in the counting process.
In summary, the number of ways to form a committee with a specified number of men and women is calculated by multiplying the combinations of men and women chosen from their respective groups.
