In algebra and other branches of mathematics, **Pascal’s triangle** is a triangular array of numbers that lists the coefficients of the expansion of any binomial expression (x + y)^{n}, where n is any positive integer and x and y are real numbers. Its construction is simple: the numbers in each row are the sum of the numbers in the preceding row. So, each row begins and ends with the number 1.

### How to Write Pascal’s Triangle

Pascal’s triangle is easy to make. Each line has one more number than the preceding line.

- Write the number 1. This is row 0.
- The numbers in each subsequent row are the sum of the numbers directly above it. For the second row, the only number above it is 1, so this row is 1, 1.
- The next row is 1, 1+2 = 3, 2+1=3, 1.
- The pattern continues infinitely.

Note that the first 1 is called the 0th row. Counting columns from the bottom, start with the 0th column, 1st column, 2nd column, etc.

### Formula for Pascal’s Triangle (Pascal’s Rule)

Alternatively, use the formula for Pascal’s triangle to find the entry for any row *n *and column *k*:

This is the same as *n*!/[*k*!(*n*–*k*)!]

Here, *n* is an integer and 0 *≤* *k* *≤* *n*. The formula is also called **Pascal’s rule**.

For example, the number in the 5th row (n) that is the 4th across (k) is 5.

5! / [4! (5-4)!)] = 5·4·3·2·1 / [4·3·2·1·1] = 5

### How to Use Pascal’s Triangle (Binomial Theorem)

The **binomial theorem** states that the *n*th row of Pascal’s triangle gives the coefficients of the expanded polynomial (x + y)^{n}.

For example, let’s expand (x + y)^{3} using Pascal’s triangle. The superscript gives the row of the triangle (3, in this case). Remember, the first “1” is row zero, so the entries in the 3rd row are: 1, 3, 3, 1. These entries are the coefficients in the expansion:

(x + y)^{3} = 1x^{3} + 3x^{2}y + 3xy^{2} + 1y^{3}

As another example, find the coefficient of the xy^{2} term in the expansion of (2x + y)^{3}.

The superscript is 3, so this is *n*. Go to the 3rd row (starting from zero) and the 2nd column to get the xy^{2} term. This value is “3”. However, the coefficient of the expression already has a 2 (coefficient of y is 1). So, multiply the Pascal’s triangle number by the existing coefficient (3 x 2^{1} x 1^{2}): 6xy^{2}

### History

Pascal’s triangle is named for 17th century French mathematician Blaise Pascal. Pascal wrote a treatise on triangles in his 1654 (published 1655) treatise *Traité du triangle arithmétique*. However, Pascal did not invent the triangle, so it bears different names in other countries.

- Around the 2nd century BC, Indian poet and mathematician Acharya Pingala described binomial coefficients and a simple method of generating them.
- 10th century Indian mathematician Halayudha offered the first surviving description of arranging the numbers into a triangle.
- Around the same time, Persian mathematician Al-Karaji (953-1029) also described the triangle.
- Al-Karaji’s book was lost, but Persian Omar Khayyám (1048-1131) described the triangle. So, in Iran, the array is called the
**Kayyam triangle**. - 11th century Chinese mathematician Jia Xian (1010-1070) knew of the triangle, which Yang Hui presented in the 13th century. In China, the array is called
**Yang Hui’s triangle**. - European mathematician Jordanus de Nemore introduced the triangle in his 13th century work,
*Arithmetic*. - In Italy, the triangle is called Tartaglia’s triangle for algebraist Niccolò Fontana Tartaglia, who wrote about it in 1556.
- Pascal’s treatise was published in 1655. French mathematician called the triangle after Pascal in 1708, when referencing the array in Pascal’s text.

### Pascal’s Triangle and the Fibonacci Sequence

When you left-justify Pascal’s triangle and add the entries diagonally, you get the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, …).

- 1
- 1
- 1 + 1 = 2
- 1 + 2 + 1 = 3
- 1 + 3 + 1 = 5
- 1 + 4 + 3 = 8

### Patterns

Pattern emerge when you consider the diagonals of Pascal’s triangle:

- The sum of the elements in the
*n*th row of Pascal’s triangle equal 2^{n}. - The triangle is symmetrical. In other words, the numbers on the right side are a mirror image of the numbers on the left side.
- The horizontal sums double with each row. For example, moving down the triangle, the sums of the elements in the rows are 1, 2, 4, 8, 16, …
- The leftmost diagonal consists only of 1s.
- The next diagonal lists the natural numbers (counting numbers; 1, 2, 3, 4, …).
- Another property of the second diagonal is that the square of a number is the sum of the numbers next to it and below it. For example, 3
^{2}= 3 + 6 = 9. 4^{2}= 6 + 10 = 16. - Moving on to the third diagonal, the triangle lists the triangular numbers (1, 3, 6, 10, …). These numbers represent the number of dots that form an equilateral triangle.
- The fourth diagonal gives the tetrahedral numbers.
- Each line gives the powers of 11. So, 11
^{0}(the first line) is 1, 11^{1}is 11, 11^{2}is 121. When you get to 11^{5}, the digits overlap to give 161051, but the pattern continues. - A number gives the number of possible paths available to it from the top of the triangle. For example, there are 3 ways to get to the number 3 in the table (moving left to right: 1,1,1,3; 1,1,2,3; 1,1,2,3).

### Probability

The triangle finds use in probability. For example tossing a coin three times has multiple outcomes corresponding to the third row of the triangle (1,3,3,1). There is one way of getting 3 heads, 3 ways of getting two heads and a tail, 3 ways of getting 2 tails and a head, and 1 way of getting three tails.

Let’s use Pascal’s triangle for finding the probability of getting 2 heads with 4 coin tosses.

Four tosses means you use row 4 (1,4,6,4,1). There are 2^{4} or 16 total possible outcomes. Six of them give 2 heads. The probability of getting exactly 2 heads is 6/16 or 37.5%.

### Pascal’s Triangle and the Sierpinski Triangle

The graphic you get from coloring in the odd numbers of Pascal’s triangle resembles the Sierpinski triangle, which is a fractal with an overall appearance of an equilateral triangle subdivided into smaller equilateral triangles.

### References

- Coolidge, J. L. (1949). “The story of the binomial theorem”.
*The American Mathematical Monthly*. 56(3): 147–157. doi:10.2307/2305028 - Edwards, A. W. F. (2002).
*Pascal’s Arithmetical Triangle: The Story of a Mathematical Idea*. JHU Press. - Fowler, David (January 1996). “The Binomial Coefficient Function”.
*The American Mathematical Monthly*. 103(1): 1–17. doi:10.2307/2975209 - Selin, Helaine (2008).
*Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures*. Springer Science & Business Media. ISBN 9781402045592. - Smith, Karl J. (2010).
*Nature of Mathematics*. Cengage Learning. ISBN 9780538737586. - Weisstein, Eric W. (2003).
*CRC Concise Encyclopedia of Mathematics*. ISBN 978-1-58488-347-0.