Pascal’s Triangle – Definition, Properties, Applications and More

# Pascal’s Triangle – Definition, Properties, Applications and More

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Esha Gupta
Associate Senior Executive
Updated on Nov 24, 2023 14:26 IST

Pascal’s Triangle, named after the French mathematician Blaise Pascal, has roots that stretch much further back in history. The discovery of Pascal’s Triangle wasn’t a single event but rather a gradual development over centuries across different cultures. Let us understand more!

Pascal’s Triangle is a triangular array of numbers. It is a mathematical concept that forms a triangle with numbers, where each number is the sum of the two numbers directly above it. The outer edges of the triangle are always 1. It starts with a single “1” at the top and expands outward with each additional row, making a symmetric, triangular array of binomial coefficients. It looks like this:

```           1
1   1
1   2   1
1   3   3   1
1   4   6   4   1

... and so on
```

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The mathematical formula to find the entry in the nth row and kth column (starting from 0) of Pascal’s Triangle is given by the binomial coefficient formula:

## It has the following properties:

1. Edges are 1s: The numbers on the outer edge of the triangle are all 1s.
2. Row Number: If you start numbering rows from 0, the nth number in the nth row is n.
3. Binomial Expansion: The coefficients in the binomial expansion of (a+b)n are the numbers in the nth row. For example, for (a+b)4, the coefficients are ‘1 4 6 4 1’, corresponding to a4 + 4a3b + 6a2b2 + 4ab3 + b4.
4. Combinatorial Interpretation: The entry in the nth row and kth column (starting counting from 0) of Pascal’s Triangle gives the number of ways to choose k items from n items, also known as “n choose k” or the binomial coefficient.
5. Hockey Stick Pattern: If you start from any number and follow a diagonal down and to the right, then turn to follow another diagonal down and to the left, the sum of the numbers in the first diagonal is given by the number where you turn.
6. Fibonacci Sequence: If you sum the diagonals’ numbers starting from the 1s on the outer edges, you get the Fibonacci sequence.
7. Symmetry: Each row in Pascal’s Triangle is symmetric; the numbers on the left side of the row mirror those on the right.
8. Sum of Rows: The sum of rows in the nth row is 2n. For instance, the sum of numbers in a row in 3 is 1+3+3+1 = 8, which is 23.
9. Triangles within the Triangle: Pascal’s triangle contains other number triangles like Catalan numbers and triangular numbers.
10. Connection to Sierpinski’s Triangle: When you highlight the odd numbers in Pascal’s triangle, a pattern similar to Sierpinski’s Triangle emerges, which is a fractal pattern.
11. Hexagonal Numbers: The difference between numbers along diagonals gives the sequence of hexagonal numbers.
12. Catalan Numbers: You can find Catalan numbers in Pascal’s triangle by summing entries along shallow diagonals.
13. Connection with Lucas Numbers: Similar to the Fibonacci sequence, the Lucas numbers can also be found in Pascal’s triangle but with a slightly different summation rule along the diagonals.
14. Power Sums: The sum of the kth entries in the first n rows yields power sums and can be used to find sums of sequences of numbers raised to power k.
15. Square Numbers: If you add the numbers in the diagonals parallel to the left-to-right diagonal, the sums are the square numbers.
16. Pi Approximation: There are methods to approximate Pi using certain properties of Pascal’s Triangle.
17. Multinomial Coefficients: Generalized binomial coefficients called multinomial coefficients can also be represented using Pascal’s Triangle’s extended version.
18. Magic 11s: Each row represents the digits of the powers of 11 (up to a point, after which carrying is necessary).

## Pattern :

```                       1
1   1
1   2   1
1   3   3   1
1   4   6   4   1
1   5  10  10   5   1
1   6  15  20  15   6   1
1   7  21  35  35  21   7   1
1   8  28  56  70  56  28   8   1
1   9  36  84 126 126  84  36   9   1
```

### Using C

Code in C

```//Pascal's triangle pattern using C#include <stdio.h> int main() { int rows = 10; // outer loop for rows for (int i = 0; i < rows; i++) { // inner loop 1 for leading white spaces for (int j = 0; j < rows - i; j++) { printf(" "); // Using two spaces for proper alignment } int C = 1; // coefficient initialized to 1 for each row's first position // inner loop 2 for printing numbers for (int k = 0; k <= i; k++) { printf("M", C); // Adjusted printing space for the coefficient C = C * (i - k) / (k + 1); } printf("\n"); } return 0;}Copy code```

### Using C++

Code in C++

```#include <iostream>#include <iomanip> // for std::setwusing namespace std; int main() { int rows = 10; for (int i = 0; i < rows; i++) { for (int j = 0; j < rows - i; j++) { cout << " "; // Using two spaces for proper alignment } int C = 1; // coefficient initialized to 1 for each row's first position for (int k = 0; k <= i; k++) { cout << setw(4) << C; // Adjusted printing space for the coefficient C = C * (i - k) / (k + 1); } cout << "\n"; } return 0;}Copy code```

### Using Python

Code in Python

```rows = 10 for i in range(rows): print(" " * (rows - i), end="") # Using two spaces for proper alignment C = 1 # coefficient initialized to 1 for each row's first position for k in range(i + 1): print(f"{C:4d}", end="") # Adjusted printing space for the coefficient C = C * (i - k) // (k + 1) print()Copy code```

### Using Java

Code in Java

```public class Main { public static void main(String[] args) { int rows = 10; for (int i = 0; i < rows; i++) { for (int j = 0; j < rows - i; j++) { System.out.print(" "); // Using two spaces for proper alignment } int C = 1; // coefficient initialized to 1 for each row's first position for (int k = 0; k <= i; k++) { System.out.printf("M", C); // Adjusted printing space for the coefficient C = C * (i - k) / (k + 1); } System.out.println(); } }}Copy code```

Thus, this is one way of printing Pascal’s triangle using different programming languages.

## Applications of Pascal’s Triangle

There are so many applications of this concept, out of which a few are listed below:

1. Geometry: It is used to find the number of distinct paths (without backtracking) from one corner to another in a grid.
2. Fractal geometry: Pascal’s Triangle modulo 2 is used to create a type of fractal called the Sierpinski Triangle.
3. Computer science: It helps in dynamic programming approaches, particularly in problems involving combinations and permutations.
4. Computer graphics: It is used in Bezier curves, which are used in various graphic design and drawing applications.
5. Finance: Particularly in pricing certain types of financial options, binomial methods that utilize Pascal’s triangle can be used.

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## Conclusion

Thus, Pascal’s Triangle is an amazing example of how seemingly simple can have deep and varied applications, showcasing the wonder and depth hidden within mathematics. Keep learning, keep exploring!