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How to approach grouping setups

Grouping setups overview

Grouping setups are common on Test Day. In this article, we’ll examine pure grouping setups (as opposed to mixed setups that include an ordering component), and share some tips to raise your game and boost your score.
We recommend that you learn how to approach ordering setups before you move on to grouping setups.

How do we recognize grouping setups?

There are a few ways you can recognize simple grouping setups, but the most obvious way might be that there is no ordering component to the task, and there’s only one kind of assignment that needs to be determined. Note that grouping setups, in comparison to ordering setups, have a bit more variety in how they’re presented to you.
A few examples of grouping tasks:
  • Determine which of 3 appetizers each of 7 people will eat.
  • Determine whether each of 8 birds are in the forest or not in the forest.
  • Determine in which of 2 seasons each of 10 employees will take a vacation.
  • Determine which, and how many, of 3 upgrades each of 6 cars have.
  • Determine which 5 of 7 people are selected to be on a committee.
Notice that order doesn’t matter in any aspect of the tasks above; for example, selecting some of 7 people to be on a committee doesn’t imply any kind of ordering to the selection—it’s simply that five people will be on the committee and two will not. So the “groups” in that example are the “In” group and the “Out” group.
Additionally, instead of putting elements into subgroups, the task could ask you to assign elements to different groups, and those elements could repeat—for example, in the first example above, there are three appetizers and seven people. The “groups” are now the people, and the elements being assigned to them are the appetizers. It’s possible that all seven people eat the cheesy bread, for example. It’s also possible that none of the people eat bacon-wrapped dates.
And if you see a grouping task, but you also see that order matters for some parts of the setup, then you’re looking at a mixed setup.
Ordering setups are covered in this article.
Mixed setups are covered in this article.

What should we pay attention to in the passage?

In the passage for a grouping setup, you’ll be given a certain number of elements and told that they will be grouped in some way. Sometimes, the numbers will be simple: 9 people and 3 subgroups, for example. Sometimes, however, the numbers won’t line up exactly that way; perhaps there will be 7 people for 3 groups and it’s possible that one group is empty. Always pay attention to how the number of elements could be split by group based on the restrictions provided!

What does the sketch look like for a grouping setup?

In general, it’s good to represent what’s happening in the setup with a diagram or sketch. The diagrams of grouping setups can be more varied than those of ordering setups, since grouping setups are more diverse in their tasks. We’ll show you a few different setups here, but the commonality is that elements are “grouped” into different categories.

Elements assigned to different sub-groups:
Suppose we have a task in which employees ABCDEF are all traveling, each employee to exactly one of countries X, Y, and Z. In addition, we learn that at least one employee goes to each destination. Your empty diagram (before establishing any rules or deductions) might look like this:
An initial diagram and a grouping diagram. The initial diagram is written in blue and lists the following letters. A, B, C, D, E, F. The grouping diagram includes 3 columns labeled from left to right as follows. X, Y, Z. Below each column is a corresponding line. All lines are blank.
Elements are either assigned to a group, or not assigned to that group (“In/Out”):
Suppose we have a task in which employees ABCDEF are either promoted or not promoted. Functionally, this isn’t too different from our first example, but the rules tend to be largely conditional in these types of setups—for example, “If A is not promoted, then B is promoted.” Since all employees must fall into one category or the other, we like to recommend an “In/Out” diagram.
Here are two options that you can use—the first option can always be used, whereas the second option can only be used if you know exactly how many elements are “in” and how many are “out.”
An initial diagram and two grouping diagrams. The initial diagram is written in blue and lists the following upper case letters vertically. A, B, C, D, E, F. A grouping diagram labeled 1 includes two columns labeled in, out is crossed out. A grouping diagram labeled 2 includes six horizontal lines with a vertical line after the fourth horizontal line. To the left of the vertical line is the word, in. To the right of the vertical line, the word out is crossed out.

What kind of rules are typical in grouping setups?

Since grouping involves assignment, then some of the most common rules will involve how elements are—or are not—assigned. Let’s continue with the task we made up above, in which each of employees ABCDEF will be traveling to exactly one of countries X, Y, and Z. We will likely see rules that resemble the following:
  • Exactly one employee will travel to X.
  • A will not travel to X.
  • A and B don’t travel to the same country.
  • A and B don’t travel to the same country, nor do either of them travel to X.
  • A and B travel to the same country.
  • There are fewer employees traveling to X than to Y.
  • If A travels to X, then B travels to Z.

What are some good ways to represent these typical rules, and what can I deduce from each one?

It’s a good idea to get into a habit of consistent rules notation. That doesn’t mean that any one notation is the “correct” one, though! Let’s talk through these rules, and we’ll give you one suggestion for each rule, and you can either use our suggestion or come up with your own. We’ll also let you know if there are any deductions that you can make for each rule, so that you can start to formulate deductions quickly and easily.
For these examples, we’ll imagine that the six employees ABCDEF are traveling to countries X, Y, and Z—each employee travels to only one country, and each country is traveled to by at least one employee.

Exactly one employee will travel to X.
A grouping diagram includes 3 columns labeled from left to right as follows. X, Y, Z. Below each column is a corresponding line. All lines are blank. The line under column X is double ruled.
Deductions:
  • We said that numbers are important in grouping (and mixed) setups. Always pay attention to what we can deduce from information regarding numbers!
  • If exactly one employee travels to X, and we know that at least one employee travels to Y and to Z, then that leaves us with three spots to establish.
  • If the setup gives you a maximum of employees who can travel to each country, then that restricts the numbers more (and we like that!). For example, if the maximum of employees who can visit each country is three, and we have six employees in our list, then we know that one employee travels to X, two or three travel to Y, and two or three travel to Z. In other words, the numbers will be 1:2:3 or 1:3:2.
  • If, however, the setup doesn’t give us a maximum of employees who can travel to each country, then the possibilities are more numerous. The numbers could be 1:1:4, 1:4:1, 1:2:3, 1:3:2.

A will not travel to X.
A grouping diagram includes 3 columns labeled from left to right as follows. X, Y, Z. Below each column is a corresponding line. All lines are blank. Below the line in column X, A is crossed out.
Deductions:
  • Since A doesn’t travel to X, we can deduce that A must travel to either Y or Z. If there were only two groups to pick from, then “not X” would mean that A must travel to the other group.
  • Since A can only travel to two different countries, look for opportunities in the questions to build two scenarios around A—this is a helpful move only if doing so leads to additional deductions.

A and B don’t travel to the same country.
A above B is a box crossed out.
Deductions:
  • Since this rule tells us about something that can’t happen, we can’t make any useful deductions from this rule alone.
  • What’s likely is that either:
  • 1) this rule will lead to deductions when it’s combined with another rule (for example, a rule that states, “B travels to Y” leads to the deduction that A can’t travel to Y), or
  • 2) this rule will lead to deductions when it’s combined with temporary rules in the questions (for example, a temporary rule might tell us, or allow us to determine, that the travelers to country X don’t include A or B, which would mean that one of A and B travels to Y and the other travels to Z).

A and B don’t travel to the same country, nor do either of them travel to X.
A grouping diagram includes 3 columns labeled from left to right as follows. X, Y, Z. A line corresponds to column y and column z. Entries read as follows. Column y. A, slash, B. Column z. A, slash, B.
Deductions:
  • Since A and B both don’t travel to X, and they can’t be a pair, that means that one of them travels to Y and the other travels to Z.
  • We don’t know which one goes where, but we can “reserve” a spot for each of them under countries Y and Z.
  • This allows us to account for A and B and create a placeholder for them so that they are included in any applicable counts or possibilities.

A and B travel to the same country.
A above B is a box.
Deductions:
  • Since grouping setups have a limited number of groups, you may be able to create scenarios around this AB pair.
  • For instance, you could draw three scenarios, one in which AB travels to X, to Y, and to Z. If, for example, you were also told that each country has a maximum of two visitors, this would greatly restrict where all of the other elements can go.

There are fewer employees traveling to X than to Z.
A grouping diagram and a notation. The grouping diagram includes 3 columns labeled from left to right as follows. X, Y, Z. Below columns X and Y is a line. Below column Z are two corresponding lines. All lines are blank. The notation reads as follows. X is less than Z.
Deductions:
  • We can deduce that Z must have at least two employees, so we can add a blank directly to our initial diagram.
  • Note that we’re not necessarily done with X and Z ! We don’t know if X has one employee and Z has two employees, or if X has two employees and Z has three employees, or if X has one employee and Z has three employees.

If A travels to X, then B travels to Z.
A grouping diagram and two notations. A grouping diagram includes 3 columns labeled from left to right as follows. X, Y, Z. Below each column is a corresponding line. All lines are blank. Notation 1 reads as follows. If A sub x, right arrow, B sub z. Notation 2 reads as follows. If B sub z is crossed out, right arrow, A sub x is crossed out.
Deductions:
  • We’ll want to remember to note the logically equivalent rule: if we learn that B doesn’t travel to Z, then we’ll deduce that A doesn’t travel to X.
  • That means that if we learn that B does travel to X or Y, then this rule triggers and we can deduce that A travels to either Y or Z.
  • If you have multiple conditional rules, it’s not necessary (or even recommended) to “chain” them all together the way we do in ordering setups. As long as you have all of the rules written out neatly on your page, you’ll be able to “follow the triggers” as you move through the questions.
  • In other words, if you have a rule stating that “if A is selected, then B is selected” and a separate rule that states, “if B is selected, then D is not selected”, don’t spend the time making a longer rule from the two rules. Just keep the individual rules neat and lined up.

Practice Example

Now, let’s take everything we’ve learned and apply it to an actual LSAT setup!

Beginning of reading passage.

A company organizing on-site day care consults with a group of parents composed exclusively of volunteers from among the seven employees—Felicia, Leah, Masatomo, Rochelle, Salman, Terry, and Veena—who have become parents this year. The composition of the volunteer group must be consistent with the following:
If Rochelle volunteers, then so does Masatomo.
If Masatomo volunteers, then so does Terry.
If Salman does not volunteer, then Veena volunteers.
If Rochelle does not volunteer, then Leah volunteers.
If Terry volunteers, then neither Felicia nor Veena volunteers.
End of reading passage.

What does the setup tell us?
We have seven employees, some of whom will be part of a volunteer group. Since it’s not necessarily true that all of the employees will be volunteers, then that’s a grouping scenario—either an employee volunteers, or that employee doesn’t volunteer. There is no ordering component to this task, since there’s no arrangement or sequence of any kind. Go ahead and make a basic sketch based on this information before we look at the rules!
What do the rules tell us?
Go ahead and note the rules, making sure to take the time to understand them, and to get them right the first time!
What can we deduce?
Now it’s time to make deductions! Can you use what we covered in this lesson in order to deduce what must be true beyond what’s presented to us at face value?
We’re in a really great spot to move on to the questions! With this diagram, we won’t have to use much (if any) time-consuming “trial and error”, and we'll gain points on Test Day.

Which one of the following could be a complete and accurate list of the volunteers?
(A) Felicia, Salman
(B) Masatomo, Rochelle
(C) Leah, Salman, Terry
(D) Salman, Rochelle, Veena
(E) Leah, Salman, Terry, Veena

If Veena volunteers, then which one of the following could be true?
(A) Felicia and Rochelle also volunteer.
(B) Felicia and Salman also volunteer.
(C) Leah and Masatomo also volunteer.
(D) Leah and Terry also volunteer.
(E) Salman and Terry also volunteer.

If Terry does not volunteer, then which one of the following CANNOT be true?
(A) Felicia volunteers.
(B) Leah volunteers.
(C) Rochelle volunteers.
(D) Salman volunteers.
(E) Veena volunteers.

If Masatomo volunteers, then which one of the following could be true?
(A) Felicia volunteers.
(B) Leah volunteers.
(C) Veena volunteers.
(D) Salman does not volunteer.
(E) Terry does not volunteer.

If Felicia volunteers, then which one of the following must be true?
(A) Leah volunteers.
(B) Salman volunteers.
(C) Veena does not volunteer.
(D) Exactly three of the employees volunteer.
(E) Exactly four of the employees volunteer.

Which one of the following pairs of employees is such that at least one member of the pair volunteers?
(A) Felicia and Terry
(B) Leah and Masatomo
(C) Leah and Veena
(D) Rochelle and Salman
(E) Salman and Terry

Takeaways

  • Order and arrangement don’t matter in grouping scenarios. In other words, you could list elements in any order and it wouldn’t change the dynamics of the task.
  • Numbers can be very important in grouping tasks. Pay attention to minimums and maximums (if there are any) per group, and look for ways that you can make deductions based on those numbers.
  • Know how to manage conditional rules! You’ll want to be very adept at forming the logical equivalent of conditional rules (sometimes known as the “contrapositive”).

Next steps

head over to the practice area
Then, once you've finished your diagnostics and created a practice schedule, head over to the practice area and try some grouping setups on your own!

Want to join the conversation?

  • blobby green style avatar for user dressupjong
    From rule 3 or 4 alone, isn't there still a possibility for both R & L or S & V to be volunteers? When one is absent, the other must volunteer. However, there is no requirement that they both cannot be volunteers. In other words at least one should volunteer. Please correct me if I'm wrong!
    (24 votes)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user Abishek Meenakshi Sundaram
    #2 - If Veena volunteers, how could Salman also volunteer? B, the chosen answer, isn't right in that case.

    #4 - If M volunteers, R and T would also volunteer. If R volunteers, L cannot. Therefore, B isn't the right answer again.

    Please clarify #2 and #4. Thanks!
    (19 votes)
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    • blobby green style avatar for user Sumant Kushwaha
      In question 2 , if salman does not volunteer then veena must volunteer, does not says anything as to what happens if salman volunteers. So the conclusion that if salman volunteers then veena can also volunteer is not contradicted.
      However , option B still is wrong because since Rochelle is not volunteering Leah has to.
      (5 votes)
  • starky sapling style avatar for user joannak08
    How are the "equivalent rules" on Rule 3 and Rule 4 correct? With Rule 3, Veena and Salman could both be in. Similarly, with Rule 4, Leah and Rochelle could both be in.

    Rule 3: If Salman is out, then Veena is in. The equivalent rule is that if Veena is out, then Salman is in.

    Rule 4: If Rochelle is out, then Leah is in. The equivalent rule is that if Leah is out, then Rochelle is in.

    These "equivalent rules" seem to commit the grave error we are warned against (creating rules that are not actually rules).
    (16 votes)
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  • blobby green style avatar for user Eugenia Edwards
    For the question: If Veena volunteers, then which one of the following could be true?

    How could the answer be (2) Felicia and Salmon could volunteer.

    If Veena volunteers, Salmon cannot, so how could he volunteer? Thus how can number (2) be the answer?
    (13 votes)
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  • piceratops seed style avatar for user Aleija
    Regarding Q2, it's explicitly a mistake on Khan's part. In Q3, they state outright this deduction: "That means that Salman must be in, since Veena is out (Rule 4)." However, in question 2, they claim: "Since we didn’t make any deductions about either Felicia or Salman, each employee could volunteer or not volunteer." They did not consider that Veena being in would negate Salman being in in Q2, but they did utlize that assumption in Q3. It really is as simple as "If V, then S" and "If S, then V." The writers even rest upon that assumption, thus the answer to 2 is wrong.
    (12 votes)
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    • blobby green style avatar for user Michael Dunlap
      So following this line of thinking on this questioning, I'm not understanding this statement here written on Khan.

      Rule 3: If Salman is out, then Veena is in. No remaining choices apply to this rule, since Salman is present in all of them.

      Since not all the answers have Salman, is this a typo on behalf of Khan?
      (A) Felicia, Salman
      (B) Masatomo, Rochelle
      (C) Leah, Salman, Terry
      (D) Salman, Rochelle, Veena
      (E) Leah, Salman, Terry, Veena
      (3 votes)
  • duskpin seedling style avatar for user drew.lew96
    Is there an error in the first practice problem? For "Which one of the following could be a complete and accurate list of the volunteers?," the explanation for the right and wrong answers do not make sense to me. Please let me know if I am just misunderstanding the question, answer and explanation. I thought the answer could be either A (F&S) or C (L,S,T). When looking at the fast approach to solving this problem, your explanation states that A cannot be the correct answer because of "Rule 4: If Rochelle is out, then Leah is in. We can cross off (A)." However, option A does not mention Rochelle or Leah at all... It mentions Felicia and Salaman, so I'm not sure how they arrived at that conclusion.
    (10 votes)
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  • blobby green style avatar for user BrandonK
    (We can then cross off (A), because three questions ago, it was possible for both Felicia and Terry to be out!)
    I am confused on how Felicia and Terry could both be OUT if T is out that means F is IN. Am I missing something?
    (9 votes)
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  • duskpin seed style avatar for user alyssack
    can someone explain why it's possible for both Salman and Veena to volunteer?
    (8 votes)
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  • blobby green style avatar for user mailforchriscarter
    Question 2 and it's explanation are very poorly written. We need a CLEAR explanation of why S and V could ever possibly be together given RULE 3.
    (7 votes)
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  • piceratops seed style avatar for user Adithi Adithi
    #1
    Which one of the following could be a complete and accurate list of the volunteers?
    (A) Felicia, Salman
    (B) Masatomo, Rochelle
    (C) Leah, Salman, Terry
    (D) Salman, Rochelle, Veena
    (E) Leah, Salman, Terry, Veena



    The given answer is C. But isn't it already given that If R is present, M has to be present and if M is present T has to be present? How can the answer be a "complete and accurate list of volunteers" then?
    R,M,T,S could be a complete and accurate list of volunteers right? But it's not given in any of the options.
    According to the article on Contrapositives, If A is absent when B is present, then, A cannot be present if B is present. Is that correct or have i misunderstood contrapositives?
    (7 votes)
    Default Khan Academy avatar avatar for user