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# Pascal's Triangle

Let's look at Pascal's triangle, how to construct one and its relevance in binomial expansions.

## What is Pascal's triangle?

Pascal's triangle is, as its name implies, a triangle which contains binomial coefficients. The top of the triangle starts with the single number 1, and as we move down the triangle, each row increases by one number.

### Binomial coefficients

Binomial coefficients are relevant in the context of binomial expansions. The general formula for a binomial expansion is:

In this case, the binomial coefficients are the constant terms that are written in the form. These coefficients can be found either by using this formula:

Or by using Pascal's Triangle:

An illustration of the Pascal's triangle

The diagram above shows the first 8 rows of Pascal's Triangle only, but this can be carried out until infinity. Each row corresponds to a number for n, with the first row being for n = 0.

Pascal's triangle with respective n values

## Constructing Pascal's triangle

Pascal's triangle has a specific pattern which makes it easier to construct rather than remember it by heart. As you might have noticed from the diagram above, each row starts and ends with 1 and the number of elements in each row increases by 1 each time. The number of elements (m) in each row is given by m = n + 1. So the row (n = 6) has 7 elements (1, 6, 15, 20, 15, 6, 1). An element can be found by adding together the two elements above it.

For example, for the third row (n = 2), the 2 comes from adding 1 + 1 from the row above:

Steps in constructing Pascal's Triangle

For the fourth row (n = 3), the two 3s come from adding 1 + 2 from above:

In the fourth row (n = 3) we add 1 + 3 to get 4, 3 + 3 to get 6 and 3 + 1 to get 4:

This process can be repeated as many times as needed until the row we need is reached.

### Sum of the rows in Pascal's triangle

In each row, the number obtained by summing all the elements in the row is given by . For example for row 3 (n = 2), the sum of the elements is 1 + 2 + 1 = 4 or = 4. This is useful to help us work out the sum of the elements for very big rows without having to construct Pascal's triangle For example, we know that for the 20th row (n = 19), the sum would be

### The Fibonacci sequence in Pascal's triangle

The Fibonacci series can be found in Pascal's triangle by adding numbers diagonally.

An illustration of the Fibonacci sequence

## Carrying out binomial expansion using Pascal's triangle

As mentioned before, Pascal's triangle is a helpful way to determine the binomial coefficients in a binomial expansion.

Let's look at how to expand.

First, we need to determine n, which is the exponent so in this case 5. This tells us that we will need to construct Pascal's Triangle until row 6 where n = 5. Using the method described above, we get:

An illustration on the use of Pascal's Triangle in binomial expansion

This means we will be using the binomial coefficients 1, 5, 10, 10, 5 and 1. Plugging this into the binomial formula, we get:

Which can be simplified to:

## Pascal's Triangle - Key takeaways

• Pascal's triangle can be constructed to help us find binomial coefficients.

• It starts at row 1, with n = 0 and a single element, 1.

• In each row, the number of elements increases by 1 and is given by m = n + 1, where m is the number of elements.

• Each row has a 1 on both extremes and the middle values are found by adding the numbers above.

• The sum of each row is .

• Fibonacci's sequence can be found by adding the elements diagonally.

• We can use Pascal's Triangle to find binomial coefficients and solve binomial expansions of the form .

Pascal's triangle is important because it helps us find the binomial coefficients for binomial expansion and can be used in probability theory, combinatorics and algebra.

Pascal's triangle is a series of rows in the shape of a triangle. The first row is at n=0 and each row has n+1 elements. It shows the binomial coefficients for all values of n.

The Fibonacci sequence can be found by adding the values in Pascal's triangle diagonally.

## Final Pascal's Triangle Quiz

Question

What can Pascal´s triangle help us find?

Binomial coefficients.

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Question

What is the $$n$$ number for the first row in Pascal´s triangle?

$$n=0$$.

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Question

What is the first value in Pascal's triangle?

$$1$$.

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Question

What are the values on the extremes of each row?

$$1$$.

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Question

How do you find the values of a row?

By adding together the values above it.

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Question

How many elements are in row $$n$$ of Pascal's triangle?

$$n+1$$.

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Question

What is the formula for finding the sum of each row?

$$2^n$$.

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Question

How do you find Fibonacci´s sequence in Pascal's triangle?

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Question

What are the first five values of Fibonacci's sequence?

1, 1, 2, 3, 5.

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Question

What is the sum of the elements in the 8th row ($$n=7$$) of Pascal's triangle?

$$128$$.

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Question

What are the binomial coefficients for the expansion $$(x+y)^6$$?

$$1, 6, 15, 20, 15, 6, 1$$.

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Question

What are the binomial coefficients for the expansion $$(x+2y)^4$$.

$$1, 4, 6, 4, 1$$.

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Question

How do you use Pascal's Triangle to find the Fibonacci numbers?

Sum along the diagonals.

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Question

If you are making Pascal's triangle from scratch, what three numbers do you put in a triangle first?

$$1$$ in all three spots.

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Question

If you want to find the next row of Pascal's triangle, what do you do to the elements of the previous row?

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