If Jeff has four movies, and must choose to watch either 1, 2, or 3 different movies GMAT Data Sufficiency

Question: If Jeff has four movies, and must choose to watch either 1, 2, or 3 different movies, which of the following represents a possible number of different arrangements of movies that Jeff could watch?

Statement 1 ALONE is sufficient to answer the question, but statement 2 alone is NOT sufficient.
Statement 2 ALONE is sufficient to answer the question, but statement 1 alone is NOT sufficient.
BOTH statements 1 and 2 TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient.
Each statement ALONE is sufficient to answer the question.
Statement 1 and 2 TOGETHER are NOT sufficient to answer the question.

“If Jeff has four movies, and must choose to watch either 1, 2, or 3 different movies,”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiencycomprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

Approach Solution 1:

Firstly, the given data to us is that jeff has a set of four movies. None of the two movies with him are the same, therefore he has four different movies. Now it is asked that Jeff can choose to watch 1, 2, or 3 different movies. It should be noted that it is not given how many movies he is going to watch but the possibility of the number of movies he could watch.

For every number of options, there can be different arrangements.

If Jeff chooses to watch only one movie, then he has to watch any one movie from the given set of four movies.
If Jeff chooses one movie, the total number of arrangements = 4
If Jeff chooses to watch two movies, he has to choose two movies from the given four movies one by one.
Therefore the first movie can be selected in 4 ways. Then there remain three movies. From these three movies, one movie can be selected in 3 ways.
After choosing 1st movie, there are three options available to choose for 2nd movie.
If Jeff chooses to watch two movies, the total number of arrangements = 4 * 3 = 12
If Jeff wants to watch three movies he has to choose three movies one by one from the given set of four movies. The first movie can be selected in 4 ways. Then from the remaining three, the second movie can be selected in 3 ways. After picking two movies we have two movies left.
From these two movies, we again have to pick one movie, which can be done in 2 ways.
If Jeff wants to watch three movies, the total number of arrangements = 4 * 3 * 2 = 24

Correct Answer: E

Approach Solution 2

Here we can use permutations to get the number of ways to permute r objects from n distinct objects.
Number of ways to permute r objects from n distinct objects - \(^nP_r\) = n! / ( (n-r)!)
Here n! = n*(n-1)*(n-2)*(n-3) *.... 3*2*1
In the given case we have n = 4 because Jeff has four movies.
In case 1 we have to select one movie to watch.
Number of ways to permute one movie from four movies = \(^4P_1\) = 4
In case 2, we have to permute two movies from four movies = \(^4P_2\) = 4! / ( (4-2)!)
= 4*3*2*1 /(2 *1) = 24 / 2 = 12
In case 3, we have to permute three movies from the given four movies =

\(^4P_3\) = 4!/(4-3)! = 4 *3 *2 * 1 = 24.

Correct Answer: E

Approach Solution 3:

If Jeff decides to watch only one movie, he must pick one from the available selection of four films.
If Jeff picks a single movie, there will be four configurations in all.
If Jeff decides to watch two films, he must select the two films from the available four sequentially.
Consequently, there are four options to choose the first movie. Three films are left after that. There are three different methods to choose one of these three films.
There are three possibilities for the second movie after selecting the first one.
If Jeff choose to see two films, there will be a total of 12 configurations (4 * 3).

If Jeff wants to watch three films, he must select them one at a time from the list of four films. There are four options for choosing the first movie. The second movie can then be chosen in one of three ways from the remaining three. We still have two movies after choosing two.
We once more need to choose one movie from these two, and there are two options for doing so.
The total number of arrangements if Jeff wants to watch three movies is 4 * 3 * 2 = 24.

Correct Answer: E

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If Jeff has four movies, and must choose to watch either 1, 2, or 3 different movies GMAT Data Sufficiency

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