Student Committee (500-600) - "Combinatorics"

[johnnyBuz]

I feel kind of stupid asking this but I am not seeing the "fast" way to solve this. I know how to solve this using a financial calculator with nCr function, but the explanation is not sufficient for me. Short of counting out all the possibilities, what is not clicking here for me? Am I supposed to do the factorial math?

A student committee that must consist of 5 members is to be formed from a pool of 8 candidates. How many different committees are possible?

To find the total number of possible committees, we need to determine the number of different five-person groups that can be formed from a pool of 8 candidates. We will use the anagram method to solve this combinations question. First, let's create an anagram grid and assign 8 letters in the first row, with each letter representing one of the candidates. In the second row, 5 of the candidates get assigned a Y to signify that they were chosen for a committee; the remaining 3 candidates get an N, to signify that they were not chosen:

A B C D E F G H
Y Y Y Y Y N N N

The total number of possible five-person committees that can be created from a group of 8 candidates will be equal to the number of possible anagrams that can be formed from the word YYYYYNNN = 8! / (5!3!) = 56. Therefore, there are a total of 56 possible committees.

[Bill@VeritasPrep]

The nice thing about calculating factorials is that a lot of the terms cancel out; each factorial contains every factorial below it. In this case, our total number of candidates is 8, and we're choosing 5. The formula for combinations is N!/(K!(N-K)!), so we'd have 8!/(5!3!). If we write out the factorials, we have:

(8*7*6*5*4*3*2*1)/(5*4*3*2*1*3*2*1)

We can see that we can cancel 5*4*3*2*! (or 5!) in the numerator and denominator, leaving us with:

(8*7*6)/(3*2*1)

The denominator equals 6, so we can cancel the 6 in the numerator, leaving us with a final step of:

8*7=56

[johnnyBuz]

Holy crap that was easy. Thank you Bill. I spent the last week working on verbal (improved to 83rd percentile in 2nd mock!) now I've gotta focus on math.