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Hi GMATters,
Here is a link to my discussion of Slot Method and how it can revolutionize your understanding of GMAT Combinatorics (Permutations and Combinations) questions. Why use those cumbersome formulas when there's an easier way?
Read more here: HERE.
Here's a sample for you:
If you can answer three Core Questions, you’ll be able to answer any Combinatorics question that the GMAT decides to throw at you.
Here they are:
How Many Spaces?
How Many Choices?
Does Order Matter?
Figuring out the number of Spaces, Choices, and whether Order Matters are basically the substance of the rest of this post.
If, on the other hand, you are interested in learning GMAT Combinatorics the correct way, then step inside...
Chapter 1: Factorials
The first and only place to start our discussion of Permutations and Combinations problems is with Factorials.
If we don’t get the concept of Factorials straight, the rest of this is never going to work.
The idea of a Factorial is best described in situ:
There are seven paintings to hang on a wall, and there are an unlimited number of places that we can hang them. In how many ways can the seven paintings be arranged?
That is, we have no restriction on which particular paintings we’re hanging–rather, we can just put them any damn place we please.
Let’s start with the Core Questions as discussed in the Introduction:
How Many Spaces?
This is sort of convenient, isn’t it? We don’t have any restriction on the spaces, so in this case the number of spaces is simply the number of objects: that is, seven total spaces because there are seven paintings. I guess we could theoretically say eight, nine, or 37 spaces, but there are only seven total things to count so we’re still limited back to seven.
So here are our seven spaces:
Let’s go to the next question.
How Many Choices?
This one’s probably easier on the surface. We have seven things, so that of course means that we can choose from–wait for it–seven paintings!
What this means in practice, of course, is that we can simply fill the spaces, counting down from seven. In other words, we have seven choices for the first space. We have then chosen one painting, meaning that we have six choices for the second space, etc.
7*6*5*4*3*2*1 = 7!
Note that as we work through the spaces left-to-right we multiply: seven choices for the first space times six choices for the second space times five choices for the third space, etc.
That is,
7 x 6 x 5 x 4 x 3 x 2 x 1 = 7! = 5040
(At this risk of assuming too much knowledge, I’m taking it for granted that if you’ve picked up this book you’re aware that the exclamation point is the mathematical shorthand for Factorial.)
Now, all that said, it’s possible that the multiplication business doesn’t come naturally. This is OK–you can remember a simple example here and refer to it if you get lost later on.
Let’s say that we have three main courses and four appetizers available. How many different meals can be made from these possibilities?
It might be simplest to look at a diagram here:
GMAT Combinatorics
What we see is that if we choose Main Course 1–that is, we “lock Main Course 1 into place”– we have four choices: one for each of the four appetizers. Likewise, if we lock Main Course 2 into place, we could have a further four, and the same for Main Course 3.
This, of course, gives us 3 * 4 =12 different meal possibilities. It’s no different when you have more than two spaces: you just multiply each space by the value in the successive space, and you’re good.
Now on to the third question.
Does Order Matter?
It bloody well should, should it not? That’s the point of the question, after all: we are calculating how many different orders we can put the paintings in.
Put this on your phone’s home screen if you need to: the most basic, native calculation in Combinatorics, provides a situation where Order Matters.
So that’s a useful thing to tattoo on the inside of your eyelids:
The Factorial calculation always creates a situation where Order Matters.
Everything else is just variations on this theme.
If you understand Factorials, we only have two more essential building blocks to worry about: 1) how to clip the Factorials by restricting the number of possible spaces (Permutations); 2) how to take one of those clipped Factorials and randomize the bastard (Combinations).
Onward and upward.
Enjoyed it so far? Check out more here: https://privategmattutor.london/gmat-pr ... nd-videos/
Here is a link to my discussion of Slot Method and how it can revolutionize your understanding of GMAT Combinatorics (Permutations and Combinations) questions. Why use those cumbersome formulas when there's an easier way?
Read more here: HERE.
Here's a sample for you:
If you can answer three Core Questions, you’ll be able to answer any Combinatorics question that the GMAT decides to throw at you.
Here they are:
How Many Spaces?
How Many Choices?
Does Order Matter?
Figuring out the number of Spaces, Choices, and whether Order Matters are basically the substance of the rest of this post.
If, on the other hand, you are interested in learning GMAT Combinatorics the correct way, then step inside...
Chapter 1: Factorials
The first and only place to start our discussion of Permutations and Combinations problems is with Factorials.
If we don’t get the concept of Factorials straight, the rest of this is never going to work.
The idea of a Factorial is best described in situ:
There are seven paintings to hang on a wall, and there are an unlimited number of places that we can hang them. In how many ways can the seven paintings be arranged?
That is, we have no restriction on which particular paintings we’re hanging–rather, we can just put them any damn place we please.
Let’s start with the Core Questions as discussed in the Introduction:
How Many Spaces?
This is sort of convenient, isn’t it? We don’t have any restriction on the spaces, so in this case the number of spaces is simply the number of objects: that is, seven total spaces because there are seven paintings. I guess we could theoretically say eight, nine, or 37 spaces, but there are only seven total things to count so we’re still limited back to seven.
So here are our seven spaces:
Let’s go to the next question.
How Many Choices?
This one’s probably easier on the surface. We have seven things, so that of course means that we can choose from–wait for it–seven paintings!
What this means in practice, of course, is that we can simply fill the spaces, counting down from seven. In other words, we have seven choices for the first space. We have then chosen one painting, meaning that we have six choices for the second space, etc.
7*6*5*4*3*2*1 = 7!
Note that as we work through the spaces left-to-right we multiply: seven choices for the first space times six choices for the second space times five choices for the third space, etc.
That is,
7 x 6 x 5 x 4 x 3 x 2 x 1 = 7! = 5040
(At this risk of assuming too much knowledge, I’m taking it for granted that if you’ve picked up this book you’re aware that the exclamation point is the mathematical shorthand for Factorial.)
Now, all that said, it’s possible that the multiplication business doesn’t come naturally. This is OK–you can remember a simple example here and refer to it if you get lost later on.
Let’s say that we have three main courses and four appetizers available. How many different meals can be made from these possibilities?
It might be simplest to look at a diagram here:
GMAT Combinatorics
What we see is that if we choose Main Course 1–that is, we “lock Main Course 1 into place”– we have four choices: one for each of the four appetizers. Likewise, if we lock Main Course 2 into place, we could have a further four, and the same for Main Course 3.
This, of course, gives us 3 * 4 =12 different meal possibilities. It’s no different when you have more than two spaces: you just multiply each space by the value in the successive space, and you’re good.
Now on to the third question.
Does Order Matter?
It bloody well should, should it not? That’s the point of the question, after all: we are calculating how many different orders we can put the paintings in.
Put this on your phone’s home screen if you need to: the most basic, native calculation in Combinatorics, provides a situation where Order Matters.
So that’s a useful thing to tattoo on the inside of your eyelids:
The Factorial calculation always creates a situation where Order Matters.
Everything else is just variations on this theme.
If you understand Factorials, we only have two more essential building blocks to worry about: 1) how to clip the Factorials by restricting the number of possible spaces (Permutations); 2) how to take one of those clipped Factorials and randomize the bastard (Combinations).
Onward and upward.
Enjoyed it so far? Check out more here: https://privategmattutor.london/gmat-pr ... nd-videos/
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