Stuck on a counting question? Listing possible outcomes may help.

By Brent Hanneson - December 5, 2020

At least half of all GMAT counting questions can be solved using the Fundamental Counting Principle (FCP, aka the slot method). However, it's not always easy to identify each stage (each slot) in the process. In these instances, you may gain some valuable insight by listing some of the possible outcomes.

Consider this question:

A multiple choice test consists of 8 questions, and each question has 4 answer choices (A, B, C and D). In how many different ways can the test be completed so that every question is answered?

A) 32

B) 70

C) 8⁴

D) 4⁸

E) 32⁴

If you're not sure how to proceed, ask yourself this question, "If I had unlimited time to list every possible outcome, how would I go about listing those outcomes?"

Well, one possible outcome would be AAAAAAAA, where every answer is A.

Another possible outcome would be BDBDBDBD

And a few more:

ADCAABAD

CBCBAADA

BDEADACC

At this point, we might recognize that the process of listing possible outcomes involves:

Choosing an answer (A, B, C or D) for question #1

Choosing an answer for question #2

Choosing an answer for question #3. . . and so on. 

So, those are the stages/slots.

Now start counting . . . 

There are 4 ways to answer question #1.

There are 4 ways to answer question #2.

There are 4 ways to answer question #3.

.

.

.

There are 4 ways to answer question #8.

By the FCP, the total number of outcomes = (4)(4)(4)(4)(4)(4)(4)(4) = 4^8 = D

Now consider this much harder question:

In how many different ways can 6 identical candies be distributed to 3 different children, if each child can receive from 0 to all 6 candies?

A) 18

B) 28

C) 60

D) 120

E) 216

By the way, this question is either barely within or slightly beyond the scope of the GMAT. That said, it's a manageable question if we start systematically listing possible outcomes (the key word is systematically).

I'm going to list each outcome in the following form: child A | child B | child C 

So, for example, 3|1|2 represents 3 candies for child A, 1 candy for child B, and 3 candies for child C. 

I’ll start with child A receiving all 6 candies, then I’ll list all possibilities with child A receiving 5 candies, then 4 candies, all the way to 0 candies. 

6|0|0

1 outcome when child A receives 6 candies

5|1|0

5|0|1

2 outcomes when child A receives 5 candies

4|2|0

4|1|1

4|0|2

3 outcomes when child A receives 4 candies

3|3|0

3|2|1

3|1|2

3|0|3

4 outcomes when child A receives 3 candies

At this point, is certainly seems like the total number of outcomes will equal 1+2+3+4+5+6+7 = 28

If I don’t see another way to solve the question, I’ll select B, and move on.

If you want to see a solution using counting methods, I have one here on GMAT Club.