Pascal's Triangle Formula (original) (raw)

Last Updated : 23 Jul, 2025

Pascal's Triangle is a triangular array of numbers where each number is the sum of the two numbers directly above it. The triangle starts with a "1" at the top, and each subsequent row contains the coefficients of the binomial expansion.

Here is an example of Pascal's Triangle:

The formula to calculate any element in Pascal's Triangle is based on binomial coefficients, denoted as:

\dbinom{n}{k} = \dfrac{n!}{k!(n - k)!}

Pascal's triangle is a beautiful concept of probability developed by the famous mathematician Blaise Pascal, which is used to find coefficients in the expansion of any binomial expression. It is a method to know the binomial coefficients of terms of binomial expression (x + y)n, where n can be any positive integer and x, and y are real numbers.

This triangle is used in different types of probability conditions. Here each row represents the coefficient of expansion of (x + y)n.

(x + y)0 = 1
(x + y)1 = 1x + 1y
(x + y)2 = 1x2 + 2xy + 1y2
(x + y)3 = 1x3 + 3x2y + 3xy2 + 1y3
(x + y)4 = 1x4 + 4x3y + 6x2y2 + 4xy3+1y4
(x + y)5 = 1x5 + 5x4y + 10x3y2 + 10x2y3 + 5xy4 + 1y5

Here the power of y in any expansion of (x + y)n represents the column of Pascal's Triangle.

• n represents the row of Pascal's triangle.
• Row and column are 0 indexed in Pascal's Triangle.

**Pascal's Triangle Construction

It's quite simple to make a Pascal's Triangle.

1. Start by creating the top row (the 0th row) with just the number 1.
2. In the following rows, each new number in Pascal's Triangle is the sum of the two numbers directly above it.
3. For example, to find the number in row 4, column 2:
   • Add the numbers from row 3, column 1, and row 3, column 2.
   • So, the number in row 4, column 2 is 1 + 2 = 3.

Pascal's Triangle Construction

**Properties of Pascal's Triangle

1. Each number in Pascal's Triangle is the sum of two numbers above it.
2. Numbers in a row are symmetric.
3. Each number represents a binomial coefficient.
4. Numbers on the left and right sides of the triangle are always 1.
5. nth row contains (n+1) numbers in it.

Read more: Important Facts about Pascal's Triangle

**Pascal's Triangle Other Formulas

The Pascal's Triangle formula to find the element in the n-th row and k-th column of the triangle is:

\binom{p}{q} = \binom{p-1}{q-1} + \binom{p-1}{q}

Here, 0 ≤ q ≤ p, p is a non-negative number

Alternatively, the formula to find the number in the nnn-th row and r-th column is given by:

\binom{p}{q} = \frac{p!}{q!(p-q)!}

And another recursive expression is:

\binom{p}{q} = \binom{p}{q-1} + \binom{p-1}{q-1}

**Pascal's Triangle Binomial Expansion

As we already know tascal's triangle defines the binomial coefficients of terms of binomial expression (x + y)n, So the expansion of (x + y)n is:

(x + y)n = a0xn + a1xn-1 + ......an-1xyn-1 + anyn

Practice Questions on Pascal's Triangle

**Question 1: Find the coefficient of the term x 2y **in the expansion of (x + y) 3 .
**Solution:

**Method 1:

We look at the row 3rd row of Pascal's Triangle because n is 3 and 1st column of the Pascal's Triangle because power of y is 1 in the term x2y. So the coefficient is 3.

**Method 2:

We simply apply nCr where n = 3, r = 1.
So coefficient of x2yin the expansion of (x + y)3 is 3C1 = 3

**Question 2: Find the coefficient of the term x 2 y 2 **in the expansion of (4x + 3y) 4 .
**Solution:

**Method 1:

We look at the row 4th row of Pascal's Triangle because n is 4 and 2nd column of the Pascal's Triangle because power of y is 2 in the term x2y2. So number in Pascal's Triangle is 6.

But we see that coefficient of x is 4 and y is 3 now since power of x is 2 and y is 2 in the term x2y2 so pascal Triangle number will be multiplied by 42 and 32 to find the coefficient.

Coefficient = 6 x 42 x 32 = 864

**Method 2:

We simply apply nCr where n = 4, r = 2.

So Pascal Triangle number of term x2y2 in the expansion of (4x +3y)4 is 4C2 = 6.
But we see that coefficient of x is 4 and y is 3 now since power of x is 2 and y is 2 in the term x2y2 so pascal Triangle number will be multiplied by 42 and 32 to find the coefficient.

Coefficient = 6 x 42 x 32 = 864

**Question 3: Write the 6th row of the Pascal's Triangle

**Solution:

6th row can be written as : 6C0 6C1 6C2 6C3 6C4 6C5 6C6
**1, 6, 15, 20, 15, 6, 1

**Question 4: Find the coefficient of the term x 4 **in the expansion of (2x + y) 4 .

**Solution:

**Method 1:

We look at the row 4th row of Pascal's Triangle because n is 4 and 0th column of the Pascal's Triangle because power of y is 0 in the term x4. So number in Pascal's Triangle is 1.

But we see that coefficient of x is 2 and y is 0 now since power of x is 4 and y is 0 in the term x4 so Pascal Triangle's number will be multiplied by 24 and 10 to find the coefficient.

Coefficient = 1 x 24 x 10= 16

**Method 2:

We simply apply nCr where n = 4, r = 0.

So Pascal Triangle number of term x4 in the expansion of (2x + y)4 is 4C0 = 1.
But we see that coefficient of x is 2 and y is 0 now since power of x is 4 and y is 0 in the term x4 so Pascal Triangle's number will be multiplied by 24 and 10 to find the coefficient.

Coefficient = 1

**Question 5: Find the coefficient of the term xy 2 **in the expansion of (2x + y) 3 .

**Solution:

**Method 1:

We look at the row 3rd row of Pascal's Triangle because n is 3 and 2nd column of the Pascal's Triangle because power of y is 2 in the term xy2. So number in Pascal's Triangle is 3.

But we see that coefficient of x is 2 and y is 1 now since power of x is 2 and y is 1 in the term xy2 so Pascal Triangle's number will be multiplied by 21 and 12 to find the coefficient.

Coefficient = 3 x 21 x 12 = 6

**Method 2:

We simply apply nCr where n = 3, r = 2.

So Pascal Triangle number of term xy2 in the expansion of (2x + y)3 is 3C2 = 3.

But we see that coefficient of x is 2 and y is 1 now since power of x is 2 and y is 1 in the term xy^2 so Pascal Triangle's number will be multiplied by 21 and 12 to find the coefficient.

Coefficient = 3 x 21 x 12 = 6