naabrams wrote:
I saw this problem on a practice test and answered it correctly, but my reasoning was a bit different from the answer explanation. I am concerned that my method got me the right answer out of luck, and I don't fully understand the answer explanation given, so I was hoping the community could provide some insight...
Here is the question and answer choices:
There are 7 executives, including the CEO and CFO, that are asked to form a small team of 5 members. However, the CEO and CFO may not both be assigned to the team. Given this constraint, how many ways are there to form the team?
A 6
B 10
C 11
D 21
E 28
The explanation provided:
P(7,5) -P(5,3) = [7!/(5!*(7-5))]-[5!/(3!*2!)] When I solved the problem, I interpreted it as a combination and not a permutation and solved as follows:
C(7,5) - C (5,2) = [(7*6*5*4*3)/(5*4*3*2*1)]-[(5*4)/(2*1)] = 11
So, is the explanation provided wrong, is my reasoning wrong (but lucky), or somewhere in between?
Find out the notation they use.
They use P(7, 5) to mean 7C5 = 7!/5!*2!
P(n, r) = nCr = n!/(r!(n-r)!) (or the way you write - C(n, r). Just that nCr is more natural to me.)
By the way, in your solution, what was your thought when you wrote C(5, 2)?
5C2 = 5C3 so the answer was correct but I would write 5C3 because I would say, "Let me select both the CEO and the CFO and 3 more people out of the remaining 5 in 5C3 ways."
That is what they do in the solution when they write P(5, 3) since they are using P(n, r) in place of nCr.
Of course, you could say that out of the remaining 5 people, I reject any 2 and all others go in the team. If that is how you chose to write 5C2, then there is nothing wrong.