sanaa.rizwan wrote:A researcher plans to indentify each participant in a certain medical experiment with a code consisting of either a single or a pair of distinct letters written in alphabetical order, what is the least number of letters that can be used if there are 12 participants and each participant is to receive a different code.

A.4
B.5
C.6
D.7
E.8

Another approach is to add a blank to the letters in order to account for the possibility of using just one letter for a code.

For example, consider answer choice A (4 letter).
Let's let the letters be A, B, C, D
We'll also add a "-" to represent a blank.
So, we must choose 2 characters from {A, B, C, D, -}
In how many ways can we select 2 characters?
We can use combinations to answer this. There are 5 characters, and we must select 2. This can be accomplished in 5C2 ways (=10 ways). As you can see, others have already shown those 10 possibilities.

Notice that, when we select 2 characters, there's only 1 possible code that can be created. The reason for this is that the 2 characters must be in alphabetical order. Or, in the case that a letter and a blank is selected, there's only one possible code as well.

Aside: If anyone is interested, we have a free video on calculating combinations (like 5C2) in your head: https://www.gmatprepnow.com/module/gmat-counting?id=789

Since A is not the correct answer, let's try B (5 letters)
Let's let the letters be A, B, C, D, E
Once again, we'll add a "-" to represent a blank.
So, we must choose 2 characters from {A, B, C, D, E, -}
There are 6 characters, and we must select 2. This can be accomplished in 6C2 ways (= 15 ways).
Perfect. B works.

Cheers,
Brent