Select your language

Suggested languages for you:
Log In Start studying!
StudySmarter - The all-in-one study app.
4.8 • +11k Ratings
More than 3 Million Downloads
Free
|
|

All-in-one learning app

  • Flashcards
  • NotesNotes
  • ExplanationsExplanations
  • Study Planner
  • Textbook solutions
Start studying

Pascal's Triangle

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:

Pascal's Triangle, An illustration of the Pascal's Triangle, StudySmarterAn 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, Pascal's Triangle with respective n values, StudySmarterPascal'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:

Pascal's Triangle, Steps in constructing Pascal's Triangle (a), StudySmarterSteps in constructing Pascal's Triangle

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

Pascal's Triangle, Steps in constructing Pascal's Triangle (b), StudySmarter

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

Pascal's Triangle, Steps in constructing Pascal's Triangle (c), StudySmarter

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.

Pascal's Triangle, An illustration of the Fibonacci sequence, StudySmarterAn 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:

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

Frequently Asked Questions about Pascal's Triangle

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?

Show answer

Answer

Binomial coefficients.

Show question

Question

What is the \(n\) number for the first row in Pascal´s triangle?

Show answer

Answer

\(n=0\).

Show question

Question

What is the first value in Pascal's triangle?

Show answer

Answer

\(1\).

Show question

Question

What are the values on the extremes of each row?

Show answer

Answer

\(1\).

Show question

Question

How do you find the values of a row?

Show answer

Answer

By adding together the values above it.

Show question

Question

How many elements are in row \(n\) of Pascal's triangle?

Show answer

Answer

\(n+1\).

Show question

Question

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

Show answer

Answer

\(2^n\).

Show question

Question

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

Show answer

Answer

By adding the values diagonally.

Show question

Question

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

Show answer

Answer

1, 1, 2, 3, 5.

Show question

Question

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

Show answer

Answer

\(128\).

Show question

Question

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

Show answer

Answer

\(1, 6, 15, 20, 15, 6, 1\).

Show question

Question

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

Show answer

Answer

\(1, 4, 6, 4, 1\).

Show question

Question

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

Show answer

Answer

Sum along the diagonals.

Show question

Question

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

Show answer

Answer

\(1\) in all three spots.

Show question

Question

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

Show answer

Answer

Add them.

Show question

60%

of the users don't pass the Pascal's Triangle quiz! Will you pass the quiz?

Start Quiz

Discover the right content for your subjects

No need to cheat if you have everything you need to succeed! Packed into one app!

Study Plan

Be perfectly prepared on time with an individual plan.

Quizzes

Test your knowledge with gamified quizzes.

Flashcards

Create and find flashcards in record time.

Notes

Create beautiful notes faster than ever before.

Study Sets

Have all your study materials in one place.

Documents

Upload unlimited documents and save them online.

Study Analytics

Identify your study strength and weaknesses.

Weekly Goals

Set individual study goals and earn points reaching them.

Smart Reminders

Stop procrastinating with our study reminders.

Rewards

Earn points, unlock badges and level up while studying.

Magic Marker

Create flashcards in notes completely automatically.

Smart Formatting

Create the most beautiful study materials using our templates.

Sign up to highlight and take notes. It’s 100% free.