Question

A blood bank needs 9 people to help with a blood drive. 14 people have volunteered.

Find how many different groups of 9 can be formed from the 14 volunteers.

Answer 1

Using the combination formula is still the best way, but it can also be approached a bit differently.

We can choose the first person in 14 ways, the second one in 13 ways (one has been already chosen), the third one in 12 ways (two have been already chosen, etc. until the ninth person.

Therefore, we can choose 9 people in
`14\cdot13\cdot12\cdot11\cdot10\cdot9\cdot8\cdot7\cdot6=726485760` ways.

However, this method accounts for the order of choosing, and we don't care about the order. Therefore, we have to divide the above result by the number of permutations of 9 people, which is
`9!=362880`

Therefore, the number of groups of 9 that can be formed is
`\frac{}{}`
`(726485760)/(362880)=2002`.

Answer 2

Explanation:

Out of 14 people choose 9 :

14 C 9 = 14! / (5! 9! ) = 2002 ways