Rebel or Embrace? How to Ace Permutation with Restrictions

by on December 20th, 2012

Today’s article comes courtesy of Veritas Prep GMAT instructor and BTG expert, David Newland.

How do you get a teenager to do something? Just tell them they can’t! If parents were to forbid their teenagers from eating broccoli then it would be sold by black market “broccoli pushers.”

There is one type of Quant question that teenagers would excel at – the permutation with restrictions question. Why? Because teenagers can’t stand to be told what not to do! The first thing that they do with a restriction is violate it.

Never too old to Rebel…

You are not a teenager anymore, but try this GMAT problem and see how violating the restrictions might work for you, too.

Donald is going to the Opera. He is going with six friends. Donald is a little claustrophobic and cannot have more than one person between him and the aisle. If the row is exactly six seats wide, how many different ways can Donald and his friends be seated if they are all seated in the same row?

One way to solve for a permutation with restriction is to find the number of possibilities without the restriction and then subtract the number of arrangements that are prohibited by the restriction.

In other words, quickly solve for 6!…the number of possible ways for 6 people to sit in a row of 6 chairs. That equals 720. Now as for that restriction on Donald’s choice of seat – act like a teenager and violate it! If you are told that he cannot sit in the middle then put him there and see what happens!

According to the restriction Donald can sit in the two seats on each end of the group. So he can sit in seats 1 and 2 or seats 5 and 6; each of these seats is either on the aisle or with just one person between Donald and the aisle. The two seats he cannot sit in are 3 and 4. So put him there!

If Donald is in seat 3, his other 5 friends can still be arranged in the other 5 seats in 5! ways. So that is 120 different arrangements if Donald is in seat 3. The same would be true in seat 4: 120 different arrangements there, too. So that is a total of 240 seating arrangements that are forbidden by the restriction. We found that number by violating the restriction to see what would happen.

To complete the problem we subtract 240 from 720 and get 480 different ways that Donald and his friends can sit down at the Opera …if we respect his wish to be near the aisle.

Not the Rebellious Type? Embrace the Restrictions!

Of course maybe you were that rare teenager who loved broccoli, was at home by 9PM every night, and never disobeyed your parents. If so, then take comfort, you do not have to rebel against restrictions, you can embrace them instead!

To embrace the restrictions simply do as you are told. In other words, only calculate those arrangements that are consistent with the restriction. For the problem above that would mean that you would only calculate the possible arrangements that featured Donald sitting in the “acceptable” locations – seats 1, 2, 5 or 6.

As you learned above, when you fix Donald in a particular seat then the other 5 people are allowed to sit in any of the remaining seats, so that, for example, if Donald is in seat 1, there are 5! arrangements possible. The same is true if he is in seats 2, 5 or 6 as well. So that there are 4(5!) or 4 *120 of 480 different arrangements  that respect the restriction.

Rebel or Embrace?

It is your choice as to which technique to use – rebel or embrace – when you face a permutation with restriction. Just remember that rebelling has its own special challenge: you are not solving for the answer directly but must subtract to get the final answer. The truth is that on the GMAT , as in life, sometimes it is better to rebel and sometimes it is better to just embrace the restrictions, so you should know how to do both.

Challenge: You can Rebel or Embrace Combinations, too!

The same thing works with combinations. The test writers like to tell you that two people cannot work together…perhaps this is because they seem to hate each other but are secretly in love, we will never know. What I do know is that you can use the same “rebel” and “embrace” method for combinations that you did for permutations.

Try the following program using both approaches:

The NBA will be selecting an All-Star team of 5 players from a total of 9 potential players. If LeBron James and Kobe Bryant cannot both be on the team (there is only one basketball after all) how many possible teams of five players can be created?

Which one do you prefer: to rebel or to embrace the restriction? Use the comments section below to give your answer and an explanation.


  • Rebel

    If all are selected: 9C5 Ways
    Put our 2 members in 1 team, rest 3 players can be selected in 7C3 ways

    Total: 9C5 - 7C3= 91 ways

    pls say i am right, or did i goof up?

  • Embrace-:

    If choosing james , total will be 8C4; if choosing Kobe again it will be 8C4 ways
    Total -: 8C4 + 8C4 = 112
    Minus 7C2 = 91 ways

    Is this method right, I seriously doubt?

    • If you are choosing James, James is already in the team and you can select 4 more players out of 7 (not 8 because Kobe can't be on the team) So, It'll be 7C4.

      I don't understand why you did Minus 7C2 ?

  • Embrace

    The  team can be selected in three ways.
    Case(I): Both LeBron James and Kobe Bryant are not in the team

    Team can be selected in 7C5 = 21 ways

    Case(II):  LeBron James is in the team but Kobe Bryant is not

    Team can be selected in 7C4 = 35 ways

    Case(III):  Kobe Bryant is in the team but LeBron James is not

    Team can be selected in 7C4 = 35 ways

    So, Total No. of ways = 21+35+35 = 91.

    • Very nicely explained! I personally prefer the team where BOTH Kobe and LeBron are not on the team...

    • "Case(I): Both LeBron James and Kobe Bryant are not in the team"

      This differentiates winner/loser. I would have happily selected '70' if it is given as a trap.

  • If Donald is going to the opera with six friends, then you would need seven seats, no?  So the whole first analysis is fundamentally flawed.

Ask a Question or Leave a Reply

The author Veritas Prep gets email notifications for all questions or replies to this post.

Some HTML allowed. Keep your comments above the belt or risk having them deleted. Signup for a Gravatar to have your pictures show up by your comment.