Please explain how to do/calculate/read this Sigma notation?

Question

Can anyone tell me what this notation means?

$2^n = \sum_{r=0}^n\binom{n}{r}$

That would be great, Thanks! For more Information go to Number of subsets of a set having r elements.

Answer 1

I'll go through each piece of notation in turn:

• $2^n$ is hopefully familiar notation. It means $\underbrace{2 \times 2 \times \ldots \times 2}_{n \text{ times}}$

• $\binom{n}{r}$ is a binomial coefficient. It is an abbreviation for $\dfrac{n!}{r! (n-r)!}$. Again, hopefully factorials are familiar to you.

• $\sum_{r=0}^n a_r$ means to add up the terms $a_r$ from $r=0$ to $r=n$. So it abbreviates $a_0 + a_1 + a_2 + \ldots + a_{n-1} + a_n$.

Let's do a small example of your exact question:

For $n=2$, the equation reads

$$2^2 = \sum_{r=0}^2 \binom{2}{r}$$

The left hand side is $4$, and the right hand side is

$$\binom{2}{0} + \binom{2}{1} + \binom{2}{2} = 1 + 2 + 1 = 4$$

And we see that the two sides are equal.

As an exercise, can you explicitly write out what the $n=3$ case says?

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I hope this helps ^_^