Quant question 182 OG 2018

Question

A researcher plans to identify each participant in a certain medical experiment with a code consisting of either a single letter 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

Hi Team

Could anyone help me to solve this problem using the same method taught in the module? I hardly understood the other explanations on the GMAT club.
I would really appreciate any help!

Dablu

Could anyone help me to solve this problem using the same method taught in the module? I hardly understood the other explanations on the GMAT club. I would really appreciate any help! Dablu

Apt0810 · Answer

See if you take number of letters as 4

Then single letter combinations : 4C1= 4

Combinations for pair = 4C2 = 6

Now total of both = 10< 12 so we cannot cover all the participants

Now if you take number as 5:

Combinations possible per the question requirement = 5C1+5C2 = 15

15>12, so with 5 letters we can cover all participants

Posted from my mobile device

gurudabl · Answer

Hey Apt0810 That makes sense. Thank You!

CrackverbalGMAT · Answer

Hello Dablu,

I want you to know that, on topics like P&C and Probability, you should never think about using one particular method to solve all questions. Don't limit yourself to methods, there are a lot of questions where logic is enough to help you solve a question.

This is one such question. As such, the best strategy for this question is the use of logic and elimination of options.

Let's start with option A. As per option A, the least number of letters required is 4. Let us take the letters P, Q , R and S. With these letters, we can make 4 'Single letter' codes and  4_C_2  i.e. 6 'Double letter codes'. This means that we can only create a maximum of 10 codes and hence will not be able to cater to 12 participants. Option A can be eliminated.

There might be a doubt in your mind like, "Is this not a question on permutations because we have to create codes?". This is absolutely justified. However, creating a 'Single letter' code can be done in 'n' ways - whether you use  n_p_1  or  n_c_1 . When it comes to 'double letter' codes, using  n_c_2  will be sufficient. This is because, for whatever pair of letters we select, we only have to arrange them in ALPHABETICAL order and hence, the other permutation will be invalid. So, this question represents  a curious case  of  the combinations formula giving you the number of permutations  you can make (now you see why I recommend not sticking to formula all the time).

Let's look at option B now. With 5 letters, say, P, Q, R, S and T, we can create 5 'Single letter codes' and  5_C_2  i.e. 10 'Double letter codes'. So, with 5 different letters, we can cater to 15 participants. Safe to say that we can take care of 12 of them. The least number of letters required is 5.

The correct answer option is B.

When you are dealing with questions like these where the numbers are small, I will always assume simple variables (like P, Q, R, S) and solve the question. This helps me visualise the question better; it also helps me in not missing out on cases which can happen if I try to do mental math in such questions.

Hope that helps!

Thanks, 
Arvind.

gurudabl · Answer

Thank you for the reply ArvindCrackVerbal Yes exactly, Yes this was one of the other thing that created the confusion. Great explanation Thanks!!

Bunuel · Answer

This question is discussed here: https://gmatclub.com/forum/a-researcher ... 34584.html Hope it helps.

Quant question 182 OG 2018

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