PHP Program to Print Pascal’s Triangle

Pascal’s Triangle is a mathematical construct named after the French mathematician Blaise Pascal. It is a triangular array of binomial coefficients, where each number is the sum of the two directly above it. In this article, we will explore different approaches to print Pascal’s Triangle in PHP.

Using Simple Loop

One of the simplest ways to generate Pascal’s Triangle is by using nested loops. This approach iterates through each row and column, calculating the binomial coefficient for each position.

PHP

<?php

function pascalsTriangle($rows) {
    for ($i = 0; $i < $rows; $i++) {
        $number = 1;
        for ($j = 0; $j <= $i; $j++) {
            echo $number . " ";
            $number = $number * ($i - $j) / ($j + 1);
        }
        echo "\n";
    }
}

// Driver code
$rows = 5;
pascalsTriangle($rows);

?>

Output

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

Using Recursion

An alternative approach involves using recursion to calculate the binomial coefficients. This method is less common but showcases the power of recursive functions.

PHP

<?php

function binomialCoefficient($n, $k) {
    if ($k == 0 || $k == $n) {
        return 1;
    } else {
        return binomialCoefficient($n - 1, $k - 1)
            + binomialCoefficient($n - 1, $k);
    }
}

function pascalsTriangle($rows) {
    for ($i = 0; $i < $rows; $i++) {
        for ($j = 0; $j <= $i; $j++) {
            echo binomialCoefficient($i, $j) . " ";
        }
        echo "\n";
    }
}

// Driver code
$rows = 5;
pascalsTriangle($rows);

?>

Output

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

Using Dynamic Programming

Dynamic programming can be employed to optimize the calculation of binomial coefficients and improve the overall efficiency of Pascal’s Triangle generation.

PHP

<?php

function pascalsTriangle($rows) {
    $triangle = array();

    for ($i = 0; $i < $rows; $i++) {
        $triangle[$i] = array(1);

        for ($j = 1; $j < $i; $j++) {
            $triangle[$i][$j] =
                $triangle[$i - 1][$j - 1] +
                $triangle[$i - 1][$j];
        }

        if ($i != 0) {
            $triangle[$i][$i] = 1;
        }
    }

    // Display the Triangle
    foreach ($triangle as $row) {
        echo implode(" ", $row) . "\n";
    }
}

// Driver code
$rows = 5;
pascalsTriangle($rows);

?>

Output

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