N Choose R Notation

"n choose r" notation is widely used in combinatorics, the study of the number of ways of doing something there is, and also has a close connection with Pascal's triangle. In this article, however, I focus on explaining the various ways n choose r can be written, and how to interpret and understand this notation.

Before I even attempt to explain how to interpret "n choose r" notation, it's worth mentioning that mathematicians have many ways of writing it in equations:

1. nCr

Note that this version has a few different subversions. You can have:

i) n as a superscript and r as a subscript. Or even...

ii) Both of n and r as subscripts.

2. n

r (this version is usually written with a pair of long brackets around the n and r.

3. C(n,r)

Really, it's all a matter of taste - you may have thought that opinions didn't come into the equation with mathematics, but actually mathematicians quite enjoy this sort of thing - Some stick adamantly to one notation, convinced that what they are using is "official", and spend their free time trying to convert other mathematicians to their ways, and participating in heated debates, others enjoy switching between many different ways depending on what takes their fancy or what mood they're in, others may even have a different notation for each day of the week! So, it's up to you. To avoid offending anyone, however, I'll just steer clear of this controversial topic and just write n choose r in words each time.

Basically, n choose r just means the number of ways of choosing r objects from n distinct objects. So if I have four sweets, coloured red, yellow, green and blue, and you could choose any two of them, that would be 4 choose 2. But what does 4 choose 2 actually equal? We can list the different choices you could make (called combinations) below:

R and Y

R and G

R and B

Y and G

Y and B

G and B

This gives a total of 6 different choices you could make. Notice that order does not matter. So if you picked the red then the yellow sweet, then we are counting this the same as if you picked the yellow then the red sweet.

Below are a few examples which you can check yourself:

· 4 choose 1 is 4

· 4 choose 4 is 1

· 4 choose 0 is 1

· 5 choose 3 is 10

· 6 choose 3 is 20

After trying it out a few times by listing the combinations, you should begin to feel more comfortable at using n choose r notation. However, once you have understood how to do this, there is much more fun to be had. Did you know that you can save yourself time by using the formula n!/(r!(n-r)!), which also happens to be the formula for Pascal's triangle? Enjoy finding out more about this topic in the links below!

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