# Coin Toss Possibilities in an expression

#### binomial theorem comes in handy

Okay, you've tossed a fair coin three times.

How do you usually represent the possibilities that might occur?

Something like this?

Yea that's fine, but, what if you can represent all this with *just an* algebraic expression?

## Finding the expression

Okay, let us allocate some variables.

Let's take

Possibilities:

- All Heads:
- the "cube" is because there are 3 heads

- the "cube" is because there are 3 heads
- 2 Heads, 1 Tail:
- similar notation as the above for the 3 possibilities:

- similar notation as the above for the 3 possibilities:
- 2 Tails, 1 Head:
- All tails:

Now, you take all those terms together and you get:

And if you notice carefully, that is just the expansion for:

Pretty clever, eh?

## Okay, what can you use it for?

Suppose someone asks you what are the possibilities that you will get at least 2 heads with 3 coin tosses, you can just look into the expression:

And add up the possibilities: 1+3 = 4! Boom!

Yes, this comes especially handy in such easy permutation questions.

## Wait a minute, isn't this the Binomial Expansion? YES IT IS!

In fact, you can find the possibilities of any of such coin tosses using the :Binomial Theorem!

So, the possibilities of a fair coin tossed 69 times:

This method is used to *represent* the possibilities, not anything else.

So instead of writing all of the possibilities, just write the binomial expansion, man.

Note: this won't work for dices, because they have 6 faces, not a binomial.

### What is new to this?

Well, you just learned that you can find the possibilities of things usingexpressions.

## Further Reading

- Binomial Distribution (Probability)
*How To Understand Combinations Using Multiplication*, BetterExplained

### :x binomial

### Continue Reading in Mathventures

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