Pascal's Triangle in C++

What is Pascal's Triangle

Pascal's Triangle is a famous number pattern shaped like a triangle. Every row starts and ends with 1, and each number inside is the sum of the two numbers directly above it. It is a classic programming exercise because it needs nested loops and a simple formula.

The pattern and rule

Here are the first five rows:

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

Look at row 4 (1 4 6 4 1): the 6 in the middle is 3 + 3 from the row above. Every inside number follows this "sum of the two above" rule, while the ends stay 1.

The C++ program

#include <iostream>
using namespace std;

int main() {
    int rows = 5;
    for (int i = 0; i < rows; i++) {
        int val = 1;                 // first number of each row is 1
        for (int j = 0; j <= i; j++) {
            cout << val << " ";
            val = val * (i - j) / (j + 1);   // next number
        }
        cout << endl;                 // new line after each row
    }
    return 0;
}

The outer loop i handles each row; the inner loop j prints the numbers in that row. Each row starts with val = 1, and the formula computes the next number as we go.

How the formula works

Instead of storing the previous row, we build each number from the one before it using:

val = val * (i - j) / (j + 1);

This is the binomial coefficient shortcut. Starting from 1, it produces the next value in the row directly. For row 4: start 1, then 1×4/1 = 4, then 4×3/2 = 6, then 6×2/3 = 4, then 4×1/4 = 1 — giving 1 4 6 4 1.

Dry run

Common mistakes

• Resetting val wrongly. Set val = 1 at the start of each row, not once before all rows.
• Integer division order. Write val * (i - j) / (j + 1) (multiply first) so integer division stays exact.
• Wrong inner loop limit. Row i has i + 1 numbers, so loop j from 0 to i inclusive.
• Forgetting the newline. Print endl after each row, or all numbers run together.

Summary

• Pascal's Triangle: edges are 1, each inside number is the sum of the two above.
• Use nested loops — outer for rows, inner for the numbers in a row.
• Start each row at val = 1 and use the binomial formula for the next number.
• Row i has i + 1 numbers.
• Multiply before dividing to keep integer maths exact.

Next, practise more shapes in nested loop patterns, or revise loops in C++ loops.