The binomial theorem is a method of expanding an expression that has been raised to any finite power. Using a binomial theorem, any algebraic expression $(a + b)$ with non-negative power can be expanded into a sum of the form $(a+b)^n = ^n \text{C}_0 a^nb^0 + ^n \text{C}_1 a^{n – 1}b^1 + ^n \text{C}_2 a^{n – 2}b^2 + … + ^n \text{C}_{n – 1} a^{1}b^{n – 1} + ^n \text{C}_n a^0b^n$. In this expression, each term with a numeric value called a coefficient can be obtained using Pascal’s Triangle.
Let’s understand what is Pascal’s Triangle and how to create Pascal’s Triangle.
What is Pascal’s Triangle?
Pascal’s Triangle is a kind of number pattern. Pascal’s Triangle is the triangular arrangement of numbers that gives the coefficients in the expansion of any binomial expression. The numbers are so arranged that they reflect a triangle.
A Pascal’s triangle is an arrangement of numbers in a triangular array such that the numbers at the end of each row are $1$ and the remaining numbers are the sum of the nearest two numbers in the above row. This concept is used widely in probability, combinatorics, and algebra. Pascal’s triangle is used to find the likelihood of the outcome of the toss of a coin, coefficients of binomial expansions in probability, etc.
How to Create a Pascal’s Triangle?
Pascal’s triangle is a special triangle that is named after Blaise Pascal. In this triangle, we start with a number $1$ at the top, then $1$s at both sides of the triangle until the end. The middle numbers are so filled that each number is the sum of the two numbers just above it.
The number of elements in the $n^\text{th}$ row is equal to $(n + 1)$ elements. Pascal’s triangle can be constructed by writing $1$ as the first and the last element of a row and the other elements of the row are obtained from the sum of the two consecutive elements of the previous row. Pascal’s triangle can be constructed easily by just adding the pair of successive numbers in the preceding lines and writing them in the new line.
The above figure shows Pascal’s Triangle up to the $6^{th}$ row.
You can see that the number at the top is $1$, from where the creation of Pascal’s triangle starts. This is called the $0^\text{th}$ row. Next, $1$s is added on both sides which form the $1^\text{st}$ row.
The coefficients of the binomial expansion start from the $2^\text{nd}$, which contain the numbers, $1$, $2$, and $1$. The number $2$ is obtained by adding the two $1$s in the previous row and after that $1$s are added on both sides.
Similarly, for the $3^\text{rd}$ row, the two $3$s in the middle are obtained by adding $1$, $2$ and $2$, $1$ of $2^\text{nd}$ row.
Moving on to the $4^\text{th}$ row, $4 = 1 + 3$, $6 = 3 + 3$, $4 = 3 + 1$, and then again two $1$s are added on both sides.
Pascal’s Triangle Formula
The formula to fill the number in the nth column and mth row of Pascal’s triangle we use Pascal’s triangle formula. The formula requires the knowledge of the elements in the $(n-1)^\text{th}$ row, and $(n-1)^\text{th}$ and $n^\text{th}$ columns.
The elements of the $n^\text{th}$ row of Pascal’s triangle are given by, $^n\text{C}_0$, $^n\text{C}_1$, $^n\text{C}_2$, …, $^n\text{C}_n$.
The formula for Pascal’s triangle is $^n \text{C}_m = ^{n-1} \text{C}_{m-1} + ^{n-1} \text{C}_{m}$
where
$^n \text{C}_m$ represents the $(m+1)^\text{th}$ element in the $n^\text{th}$ row
$n$ is a non-negative integer, and $0 \le m \le n$
Examples of Pascal’s Triangle Formula
Example 1: Find the value of $^4 \text{C}_2$ using Pascal’s Triangle.
To get the value of $^4 \text{C}_2$ using Pascal’s Triangle, find the $3^\text{rd}$ element in the $4^\text{th}$ row. From the triangle, we see that the value is $6$.
Therefore, $^4 \text{C}_2 = 6$.
Example 2: Find the value of $^6 \text{C}_4$ using Pascal’s Triangle.
To get the value of $^6 \text{C}_4$ using Pascal’s Triangle, find the $5^\text{rd}$ element in the $6^\text{th}$ row. From the triangle, we see that the value is $15$.
Therefore, $^6 \text{C}_4 = 15$.
Pascal’s Triangle in Binomial Expansion
Pascal’s triangle can also be used to find the coefficient of the terms in the binomial expansion $(x+y)n = a_0 x^n y^0+ a_1 x^{n-1} y^1 + a_2 x^{n-2} y^2 + … + a_n x^0 y^n$. Pascal’s triangle is a handy tool to quickly verify if the binomial expansion of the given polynomial is done correctly or not.
Examples of Pascal’s Triangle in Binomial Expansion
Example 1: Expand $(x+y)^2$ using Pascal’s triangle.
Since the power in $(x+y)^2$ is $2$, therefore, we will look into the $2^\text{nd}$ row of Pascal’s triangle. The numbers in the second row of Pascal’s triangle are $1$, $2$, and $1$. These numbers will be the coefficients of the terms in the expansion of $(x+y)^2$.
Therefore, $(x+y)^2 = 1 \times x^2 \times y^0+ 2 \times x^1 \times y^1 + 1 \times x^0 \times y^2 = x^2 + 2xy + y^2$.
Example 2: Expand $(x+y)^5$ using Pascal’s triangle.
Since the power in $(x+y)^5$ is $5$, therefore, we will look into the $5^\text{th}$ row of Pascal’s triangle. The numbers in the second row of Pascal’s triangle are $1$, $5$, $10$, $10$, $5$, and $1$. These numbers will be the coefficients of the terms in the expansion of $(x+y)^5$.
Therefore, $(x+y)^5 = 1 \times x^5 \times y^0 + 5 \times x^4 \times y^1 + 10 \times x^3 \times y^2 + 10 \times x^2 \times y^3 + 5 \times x^1 \times y^4 + 1 \times x^0 \times y^5$
$ = x^5 + 5x^4y + 10x^3y^2 + 10x^2y^3 + 5xy^4 + y^5$.
Pascal’s Triangle in Probability
Pascal’s triangle can be used in various places in the field of mathematics. Pascal’s triangle is used in probability and can be used to find the number of combinations, etc. It gives us the number of combinations of heads or tails that are possible from the number of tosses.
For example, if we toss a coin two times, we get $\text{HH}$ $1$ time, $\text{HT}$ or $\text{TH}$ $2$ times, and $\text{TT}$ $1$ time, which is the exact match of the elements in the second row of Pascal’s triangle.
The following table shows the results in the various number of tosses.
| Rows in Pascal’s Triangle(or Number of Tosses) | Outcomes in Combinations | Numbers in Pascal’s Triangle |
| $1$ | $\{\text{H}\}$, $\{\text{T}\}$ | $1$, $1$ |
| $2$ | $\{\text{HH} \}$, $\{\text{HT}, \text{TH} \}$, $\{\text{TT} \}$ | $1$, $2$, $1$ |
| $3$ | $\{\text{HHH} \}$, $\{\text{HHT}, \text{HTH}, \text{THH} \}$, $\{\text{HTT}, \text{THT}, \text{TTH} \}$, $\{\text{TTT} \}$ | $1$, $3$, $3$, $1$ |
| $4$ | $\{\text{HHHH} \}$, $\{\text{HHHT}, \text{HHTH}, \text{HTHH}, \text{THHH} \}$, $\{\text{HHTT}, \text{HTHT}, \text{HTTH}, \text{THHT}, \text{THTH}, \text{TTHH} \}$, $\{\text{HTTT}, \text{THTT}, \text{TTHT}, \text{TTTH} \}$, $\{\text{TTTT} \}$ | $1$, $4$, $6$, $4$, $1$ |
Fibonacci Series in Pascal’s Triangle
By adding the numbers in the diagonals of Pascal’s triangle, we obtain the Fibonacci sequence.
There are various ways to show the Fibonacci numbers on Pascal’s triangle. R. Knott was able to find the Fibonacci appearing as sums of “rows” in Pascal’s triangle. He moved all the rows over by one place and here the sums of the columns would represent the Fibonacci numbers.
Properties of Pascal’s Triangle
The following are some of the important properties of Pascal’s triangle.
- Each number is the sum of the two numbers above it
- The outside numbers are all $1$
- The triangle is symmetric
- The first diagonal shows the counting numbers
- The sums of the rows give the powers of $2$
- Each row gives the digits of the powers of $11$
- Each entry is an appropriate “combinatorics number” and is the “binomial coefficient”
- The Fibonacci numbers are there along diagonals
Practice Problems
- Find the following numbers in Pascal’s triangle
- $2^\text{nd}$ row, $1^\text{st}$ column
- $4^\text{th}$ row, $3^\text{rd}$ column
- $6^\text{th}$ row, $4^\text{th}$ column
- $10^\text{th}$ row, $8^\text{th}$ column
- Expand the following using Pascal’s triangle
- $(x + y)^4$
- $(x + y)^7$
- $(2x – 3y)^6$
FAQs
What is Pascal’s Triangle?
Pascal’s triangle is a special triangle that is named after Blaise Pascal. In this triangle, we start with a number $1$ at the top, then $1$s at both sides of the triangle until the end. The middle numbers are so filled that each number is the sum of the two numbers just above it.
What is Pascal’s Triangle rule?
In this triangle, we start with a number $1$ at the top, then $1$s at both sides of the triangle until the end. The middle numbers are so filled that each number is the sum of the two numbers just above it.
Why is Pascal’s triangle important?
The importance of Pascal’s Triangle lies in the fact that it is widely used in probability, combinatorics, and algebra.
Why is it called Pascal’s triangle?
Pascal’s triangle is named after the famous 17th-century French mathematician Blaise Pascal because of his work on many triangle properties.
How do you use Pascal’s triangle in a binomial expansion?
Pascal’s triangle can also be used to find the coefficient of the terms in the binomial expansion $(x+y)n = a_0 x^n y^0+ a_1 x^{n-1} y^1 + a_2 x^{n-2} y^2 + … + a_n x^0 y^n$. The numbers in each row of Pascal’s triangle correspond to coefficients in the expansion of binomial expression.
What is Pascal’s triangle formula?
The formula for Pascal’s triangle is $^n \text{C}_m = ^{n-1} \text{C}_{m-1} + ^{n-1} \text{C}_{m}$
where
$^n \text{C}_m$ represents the $(m+1)^\text{th}$ element in the $n^\text{th}$ row
$n$ is a non-negative integer, and $0 \le m \le n$
What is the first element in each row of Pascal’s triangle?
The first, as well as the last element of each row of Pascal’s triangle, is $1$.
Conclusion
Pascal’s triangle is a special triangle that is named after Blaise Pascal. In this triangle, we start with a number $1$ at the top, then $1$s at both sides of the triangle until the end. The middle numbers are so filled that each number is the sum of the two numbers just above it. Pascal’s Triangle is used widely in probability, combinatorics, and algebra.
Recommended Reading
- What is Binomial Theorem – Formula, Expansion & Examples
- What is Combination – Definition, Types, Formulas & Examples
- What is Permutation – Definition, Types, Formulas & Examples
- What is Factorial of a Number – Definition, Formula & Examples
- Fundamental Principle of Counting – Rules, Formulas & Examples
- Why Is 0 Factorial 1?
