Counting methods

manelgirona wrote:A doctor must label 12 patients. Each label contains either a single letter or a pair of different letters in alphabetical order. What is the minimum number of letters needed?

a) 4
b) 5
c) 6
d) 10
e) 12

Answer is B

My guess is that trial and error is the quickest way to solve this problem.

Let's start with (a), since we want the minimum number of letters:

4 - let's choose ABCD.

So, our labels are:
A
B
C
D
AB
AC
AD
BC
BD
CD

that's only 10, so not enough.

Now, we're only 2 shy of what we need and adding another letter will add far more than just 2 more possible labels, so pick (B), no need to test.

If for some reason you really want a formula, it's:

n + nC2

in which n is the number of letters.

n gives us the single labels and nC2 gives us the double labels (since they have to be alphabetical we use combinations instead of permutations).

So, for n=4 we have 4 + 4C2 = 4 + 6 = 10 labels

for n=5 we have 5 + 5C2 = 5 + 10 = 15 labels

Stuart Kovinsky wrote:If for some reason you really want a formula, it's:

n + nC2

in which n is the number of letters.

n gives us the single labels and nC2 gives us the double labels (since they have to be alphabetical we use combinations instead of permutations).

So, for n=4 we have 4 + 4C2 = 4 + 6 = 10 labels

for n=5 we have 5 + 5C2 = 5 + 10 = 15 labels

second method seems to be methodical and I think would help us more in genralised way ...for other problems...

shashank.ism wrote:
second method seems to be methodical and I think would help us more in genralised way ...for other problems...

I actually disagree.

While it's good to be familiar with counting methods (especially the permutations and combinations formulas), this is the first question I've ever seen that required that exact formula.

On the other hand, I've seen thousands (no exaggeration there - I mean that literally) of questions that are amenable to working with the answer choices and/or trial and error.

So, in the long run, it's far more valuable for the GMAT to understand how to work with the choices and use non-textbook approaches than it is to memorize every iteration of every possible formula that could appear.