How is choosing a boy and a girl from 12 boys and 12 girls to represent a club different from choosing two girls from 12 girls to be president and treasurer of the club?
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the first involves 24 choices. the other involve 12 choices.
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When considering choosing a boy and a girl from 12 boys and 12 girls there are 12 choices of girl for each boy selected. So the total number of ways of choosing is:\[12\times 12\]
When choosing 2 girls from 12 to be president and treasurer of the club there are 2 ways of assigning the offices for each pair chosen. So in this case the number of ways of choosing is:
What does the exclamation point mean?
The exclamation point indicates a factorial. A factorial is the product of all the positive integers from 1 up to and including a given integer. The symbol is n! where n is the given integer.
So 5 factorial is written 5! and means
\[5\times 4\times 3\times 2\times 1(=120)\]
Also 12P2 means the number of permutations of 12 different things taken 2 at a time.
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You can look at the choice of president and treasurer from the 12 girls very simply as follows:
If one of the 12 girls is chosen as president there are 11 choices for treasurer. So for each one of the girls chosen as president there are 11 choices for treasurer. Going through each in sequence it will be seen that each one if the girls has a possibility of being either president or treasurer and the total number of ways of choosing is