Please help with the combinations and Permutations problem below

Question

A club has 30 members under the age of 30 and 40 members who are 30 or older. In how many ways can a slate of officers be chosen if the President and atleast one other officer must be at least 30 years old?

Side note: the slate of officers is President, VP, Secretary, treasure, Press, and math consultant (this does not affect the problem)

Please help, Im studying for my exam and something like this might be on it

Answer 1

There are a total of 70 members in the club and 6 offices available.  This is not a permutation since order is not relevant.  Remember that always: if order is not important - it's a combination.  If it says arrangement or order, then you have a permutation.

 

Now the Pres and one other officer (it doesn't matter, so say math consultant) must be at least 30, so they are chosen from the group of 30 or older: C(40,2) because there are 40 of them to choose from.  (I'm assuming you know how to do the combination formula).

 

The other four officers can be chosen from the rest of the 68 members left (this is where you have to be careful, there is nothing saying the rest cannot be 30 or older).  We subtract two because we just chose two officers.  So there are C(68,4) ways to select the remaining 4 officers.

 

(note: words like choose, select, etc. signal the use of combination).

 

Now what do you do with these two numbers.  Of course, Counting Principle: multiply them.

 

C(40,2) x C(68,4).

 

Good luck.