In the realm of Data Structures and Algorithms (DSA), permutations and combinations play a crucial role. They are fundamental concepts in combinatorics and have wide-ranging applications in computer science, particularly in DSA.

## Permutations

A permutation is an arrangement of all the members of a set into some sequence or order. The number of permutations on a set of `n`

elements is given by `n!`

(n factorial).

Here’s a simple JavaScript function to calculate `n!`

:

JavaScript

```
function factorial(n) {
if (n === 0) {
return 1;
}
return n * factorial(n - 1);
}
```

## Combinations

A combination is a selection of items from a larger set, where the order of selection does not matter. The number of combinations of `n`

items taken `r`

at a time is given by `nCr = n! / r!(n-r)!`

.

Here’s a JavaScript function to calculate `nCr`

:

JavaScript

```
function combination(n, r) {
return factorial(n) / (factorial(r) * factorial(n - r));
}
```

## Top 5 Algorithms for Implementing Permutations and Combinations

### 1. Generating All Permutations

Here’s a simple recursive algorithm to generate all permutations of a given string:

JavaScript

```
function permute(str, l, r) {
if (l == r)
console.log(str);
else {
for (let i = l; i <= r; i++) {
str = swap(str, l, i);
permute(str, l + 1, r);
str = swap(str, l, i); // backtrack
}
}
}
function swap(str, i, j) {
let charArray = str.split('');
let temp = charArray[i];
charArray[i] = charArray[j];
charArray[j] = temp;
return charArray.join('');
}
```

### 2. Generating All Combinations

Here’s a simple recursive algorithm to generate all combinations of a given string:

JavaScript

```
function combine(str, start, end, output) {
for (let i = start; i <= end; i++) {
output += str[i];
console.log(output);
if (i < str.length - 1) {
combine(str, i + 1, end, output);
}
output = output.slice(0, output.length - 1);
}
}
```

### 3. Heap’s Algorithm for Generating Permutations

Heap’s algorithm generates all possible permutations of `n`

objects. It minimizes movement: it generates each permutation from the previous one by interchanging a single pair of elements.

JavaScript

```
function heapPermutation(a, size, n) {
if (size == 1) {
console.log(a.join(''));
return;
}
for (let i = 0; i < size; i++) {
heapPermutation(a, size - 1, n);
if (size % 2 == 1) {
let temp = a[0];
a[0] = a[size - 1];
a[size - 1] = temp;
} else {
let temp = a[i];
a[i] = a[size - 1];
a[size - 1] = temp;
}
}
}
```

### 4. Lexicographically Sorted Permutations

This algorithm generates permutations in lexicographically sorted order.

JavaScript

```
function sortedPermutations(str) {
let chars = str.split('').sort();
while (true) {
console.log(chars.join(''));
let i = chars.length - 2;
while (i >= 0 && chars[i] >= chars[i + 1]) {
i--;
}
if (i < 0) {
break;
}
let j = chars.length - 1;
while (chars[j] <= chars[i]) {
j--;
}
[chars[i], chars[j]] = [chars[j], chars[i]]; // swap
chars = chars.slice(0, i + 1).concat(chars.slice(i + 1).reverse());
}
}
```

### 5. Combinations using Backtracking

This algorithm generates all combinations of a string using backtracking.

JavaScript

```
function combinations(str, index, data, i) {
if (index == i) {
console.log(data.join(''));
return;
}
if (i >= str.length)
return;
data[index] = str[i];
combinations(str, index + 1, data, i + 1);
combinations(str, index, data, i + 1);
}
```

These are just a few examples of how permutations and combinations can be implemented in JavaScript. They are fundamental concepts in DSA and understanding them can help solve complex problems more efficiently.

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