As a continuation of my introduction to Combinations and Permutations, here are some practice problems to test the strategies I demonstrated. Also, I am trying out a new format. Instead of giving you practice problems and then writing out a half-page description for each one, I am simply going to give you the questions. But don’t worry – I am not leaving you in the lurch. For explanations, I am solving each of the questions – using my special approach, of course – in the video below.

Questions:

1. A committee is composed of a president, a vice president, and a treasurer. If six people are trying out for the three positions, how many different committees result?

(A) 20 (B) 40 (C) 60 (D) 105 (E) 120

2. A committee of three is to be chosen from six. How many unique committees result?

(A) 20 (B) 40 (C) 60 (D) 105 (E) 120

3. A committee is composed of a president, a vice president, and a treasurer. If five people are running for president, six people are running for vice president, and three are running for treasurer, how many unique committees result?

(A) 15 (B) 45 (C) 75 (D) 90 (E) 120

4. A jousting tournament requires that a team consist of two knights and two squires. The Merry Band is forming a team from five knights and three squires. How many different lineups can The Merry Band field?

(A) 10 (B) 13 (C) 15 (D) 30 (E) 120

5. A septet, a group composed of seven players, is made up of four strings and three woodwind instruments. If seven students try out for strings and seven different students try out for woodwinds, how many unique septets can result?

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You should be able to see the video explanations for these questions on this blog post. If you’re having issues viewing the videos, I’d recommend trying a different browser. You can also watch the videos on our YouTube channel. Here are the links to the explanations on YouTube:

Five years after this blog has been posted and it’s still making combinations and permutations a less threatening topic for students. Someone came to me for help on this and I didn’t at all recognize the name of the topic, so I started looking it up and searching for some practice problems and came across this website. Now, I look forward to teaching this method and helping out the way you have. Thank you so much Chris for this fantastic post!

I just wanted to make sure, this technique is only applicable for problems where you can’t repeat items right? Is there another technique similar to this for combo and permutation problems that allow you to repeat?

Wow, I have been trying to understand these types of problems for weeks via the “unmentionable other GRE study book” method and just spending 10 minutes watching you explain and it all makes perfect sense. Thank you so much!

The difference between 1 and 2 is subtle. Imagine that the president, vice-president and treasurer will stand on podiums of different height, much the way the gold, silver, and bronze medalists do at the Olympics. Being chosen as three people from the six doesn’t tell you on which podium they are standing. So once we’ve chosen three people from the six, we have to come up with a way to figure out the number of different ways the three can stand on the podiums.

The second example, by contrast, doesn’t focus on doing anything different with the members once they are in the group. In other words, there are no podiums or designations (president, vice-president, etc.). Once the three people are in the group, they are in the group. We don’t care where they stand, sit, etc.

So is it safe to assume that any problem that asks “how many unique combinations” will be a Combinations problem? If so, are there any other ‘key’ words that we can use to identify P vs C?

Well…I don’t think most combination problems would explicitly state “how many unique combinations”. Also, I can imagine a question with a lock and, say, 3 different slots with single digit numbers. How many unique combinations on the lock? This is not a combinations problem.

Typically, when the question asks about group/teams/lineups, then it is a combination problem. As long, that is, as the question is not asking about unique positions within the group/team/lineup (e.g., 1st place, 2nd place, etc.; president, vice-president, etc.).

Thanks Chris! your blogs are really very helpful when approaching perm/comb questions. I hated this topic as I had nvr really seen it before GRE prep, but now at least I can approach them w/ somewhat understanding! I used to look at bunch of formulas and get really confused. Thanks to your videos and blogs.

Great! I’m happy I was able to demystify this pesky question type. Once you get used to the dash method you don’t even have to worry about the formidable formula :)!

Before viewing your blog along with these videos, I was disconcerted and almost ready to give up when it came to trying to solve these combination-permutation problems; but now I can firmly say that I am confident when approaching these questions on the practice GRE tests. Thanks Chris!

Great! I always love to hear comments like these. Most books out there overly complicate combinations and permutations. So I’m happy I elucidated instead of obfuscated :).

Oh my God! lifesaver… advanced pre-calc. test tomorrow, and i was literally tearing up because I was so frustrated. I finally understood this, and used this as a test to see how well I knew it. Needless to say, the dash method is my backup to see if I did it right. Thanks! you’re great!

I know how confusing and frustrating this topic can be…I am happy the video made things easier. Indeed over the years I’ve turned many students into combinations/permutations fans. One student started doing them for fun once I showed him this method!

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Hello Chris,

What are the answers to questions 3 , 4 and 5?

Hi Vatsal ๐

You should be able to see the video explanations for these questions on this blog post. If you’re having issues viewing the videos, I’d recommend trying a different browser. You can also watch the videos on our YouTube channel. Here are the links to the explanations on YouTube:

* Question 3 (starting at about 1:54)

* Question 4

* Question 5

Happy studying ๐

Five years after this blog has been posted and it’s still making combinations and permutations a less threatening topic for students. Someone came to me for help on this and I didn’t at all recognize the name of the topic, so I started looking it up and searching for some practice problems and came across this website. Now, I look forward to teaching this method and helping out the way you have. Thank you so much Chris for this fantastic post!

We are so glad to hear that this post is helpful, Sara! ๐

These are super helpful!

I just wanted to make sure, this technique is only applicable for problems where you can’t repeat items right? Is there another technique similar to this for combo and permutation problems that allow you to repeat?

Thank you so much for these practice problems and tips! They are so helpful, and I would be so confused without this website.

Hi Lauren,

I’m glad you’ve found the tips helpful!

Wow, I have been trying to understand these types of problems for weeks via the “unmentionable other GRE study book” method and just spending 10 minutes watching you explain and it all makes perfect sense. Thank you so much!

Very very helpful, now i know how to identify between permutation and a combination.

Great! It’s a very tough concept to wrap one’s head around, so I’m happy you got it :).

Hi Chris,

Could you please explain the difference between questions 1. and 2. Both the scenarios seem the same to me!

Thanks very much.

The difference between 1 and 2 is subtle. Imagine that the president, vice-president and treasurer will stand on podiums of different height, much the way the gold, silver, and bronze medalists do at the Olympics. Being chosen as three people from the six doesn’t tell you on which podium they are standing. So once we’ve chosen three people from the six, we have to come up with a way to figure out the number of different ways the three can stand on the podiums.

The second example, by contrast, doesn’t focus on doing anything different with the members once they are in the group. In other words, there are no podiums or designations (president, vice-president, etc.). Once the three people are in the group, they are in the group. We don’t care where they stand, sit, etc.

Hope that helps!

Excellent! The analogy really makes it clear. Thank you!

You are welcome!

The analogy really makes it very clear. Thank you!

Hi Chris, my maths sucks! I found your explanations really really helpful!

Thanks, I’m happy they were able to help you get over the hump :). Keep it up!

So is it safe to assume that any problem that asks “how many unique combinations” will be a Combinations problem? If so, are there any other ‘key’ words that we can use to identify P vs C?

Thanks this is all really helpful!

Well…I don’t think most combination problems would explicitly state “how many unique combinations”. Also, I can imagine a question with a lock and, say, 3 different slots with single digit numbers. How many unique combinations on the lock? This is not a combinations problem.

Typically, when the question asks about group/teams/lineups, then it is a combination problem. As long, that is, as the question is not asking about unique positions within the group/team/lineup (e.g., 1st place, 2nd place, etc.; president, vice-president, etc.).

Hope that helps!

Thanks Chris! your blogs are really very helpful when approaching perm/comb questions. I hated this topic as I had nvr really seen it before GRE prep, but now at least I can approach them w/ somewhat understanding! I used to look at bunch of formulas and get really confused. Thanks to your videos and blogs.

Hi Shamila,

Great! I’m happy I was able to demystify this pesky question type. Once you get used to the dash method you don’t even have to worry about the formidable formula :)!

Before viewing your blog along with these videos, I was disconcerted and almost ready to give up when it came to trying to solve these combination-permutation problems; but now I can firmly say that I am confident when approaching these questions on the practice GRE tests. Thanks Chris!

Great! I always love to hear comments like these. Most books out there overly complicate combinations and permutations. So I’m happy I elucidated instead of obfuscated :).

awesome method ..just tempted to solve many problems on this topic

Great! The method definitely makes thing go much faster!

Oh my God! lifesaver… advanced pre-calc. test tomorrow, and i was literally tearing up because I was so frustrated. I finally understood this, and used this as a test to see how well I knew it. Needless to say, the dash method is my backup to see if I did it right. Thanks! you’re great!

Nora,

You are welcome :). Good luck on the test!

thanks! I thought it was super hard but thats because of the geometric series stuff.. i ended up using your method for all of it..:)

Great – I am happy to hear my method was helpful!

Thank you sooo much, it was very easy to comprehend! I’ve always dreaded these and you made them easier.

You’re welcome! I am always happy to hear that from students, because permutations/combinations doesn’t need to be intimidating.

your videos are really helpful,thank u…….

Great! I am happy they were helpful!

very very helpful. thank u sooooooo much

Thanks Zafran!

I know how confusing and frustrating this topic can be…I am happy the video made things easier. Indeed over the years I’ve turned many students into combinations/permutations fans. One student started doing them for fun once I showed him this method!