---
id: 5a23c84252665b21eecc7ee0
title: Left factorials
challengeType: 5
---

## Description
<section id='description'>
<b>Left factorials</b>,  $ !n $,  may refer to either  <i>subfactorials</i>  or to  <i>factorial sums</i>. The same notation can be confusingly seen used for the two different definitions. Sometimes,  <i>subfactorials</i>  (also known as <i>derangements</i>)  may use any of the notations:
<ul>
<li>$!n`$</li>
<li>$!n$</li>
<li>$n¡$</li>
</ul>
(It may not be visually obvious, but the last example uses an upside-down exclamation mark.) This task will be using this formula for <b>left factorial</b>:
$ !n = \sum_{k=0}^{n-1} k! $
where $!0 = 0$
</section>

## Instructions
<section id='instructions'>
Write a function to calculate the left factorial of a given number.
</section>

## Tests
<section id='tests'>

``` yml
tests:
  - text: <code>leftFactorial</code> should be a function.
    testString: assert(typeof leftFactorial == 'function', '<code>leftFactorial</code> should be a function.');
  - text: <code>leftFactorial(0)</code> should return a number.
    testString: assert(typeof leftFactorial(0) == 'number', '<code>leftFactorial(0)</code> should return a number.');
  - text: <code>leftFactorial(0)</code> should return <code>0</code>.
    testString: assert.equal(leftFactorial(0), 0, '<code>leftFactorial(0)</code> should return <code>0</code>.');
  - text: <code>leftFactorial(1)</code> should return <code>1</code>.
    testString: assert.equal(leftFactorial(1), 1, '<code>leftFactorial(1)</code> should return <code>1</code>.');
  - text: <code>leftFactorial(2)</code> should return <code>2</code>.
    testString: assert.equal(leftFactorial(2), 2, '<code>leftFactorial(2)</code> should return <code>2</code>.');
  - text: <code>leftFactorial(3)</code> should return <code>4</code>.
    testString: assert.equal(leftFactorial(3), 4, '<code>leftFactorial(3)</code> should return <code>4</code>.');
  - text: <code>leftFactorial(10)</code> should return <code>409114</code>.
    testString: assert.equal(leftFactorial(10), 409114, '<code>leftFactorial(10)</code> should return <code>409114</code>.');
  - text: <code>leftFactorial(17)</code> should return <code>22324392524314</code>.
    testString: assert.equal(leftFactorial(17), 22324392524314, '<code>leftFactorial(17)</code> should return <code>22324392524314</code>.');
  - text: <code>leftFactorial(19)</code> should return <code>6780385526348314</code>.
    testString: assert.equal(leftFactorial(19), 6780385526348314, '<code>leftFactorial(19)</code> should return <code>6780385526348314</code>.');
```

</section>

## Challenge Seed
<section id='challengeSeed'>
<div id='js-seed'>

```js
function leftFactorial(n) {
  // Good luck!
}
```

</div>
</section>

## Solution
<section id='solution'>

```js
function leftFactorial(n) {
    if (n == 0)
        return 0
    if (n == 1)
        return 1;

    // Note: for n>=20, the result may not be correct.
    // This is because JavaScript uses 53 bit integers and
    // for n>=20 result becomes too large.

    let res = 2, fact = 2;
    for (var i = 2; i < n; i++) {
        res += fact;
        fact *= (i + 1);
    }

    return res;
}
```

</section>