---
id: 5900f51a1000cf542c51002d
title: 'Problem 430: Range flips'
challengeType: 5
forumTopicId: 302101
dashedName: problem-430-range-flips
---

# --description--

$N$ disks are placed in a row, indexed 1 to $N$ from left to right.

Each disk has a black side and white side. Initially all disks show their white side.

At each turn, two, not necessarily distinct, integers $A$ and $B$ between 1 and $N$ (inclusive) are chosen uniformly at random. All disks with an index from $A$ to $B$ (inclusive) are flipped.

The following example shows the case $N = 8$. At the first turn $A = 5$ and $B = 2$, and at the second turn $A = 4$ and $B = 6$.

<img class="img-responsive center-block" alt="example for N = 8, with first turn A = 5 and B = 2, and second turn A = 4 and B = 6" src="https://cdn.freecodecamp.org/curriculum/project-euler/range-flips.gif" style="background-color: white; padding: 10px;">

Let $E(N, M)$ be the expected number of disks that show their white side after $M$ turns. We can verify that $E(3, 1) = \frac{10}{9}$, $E(3, 2) = \frac{5}{3}$, $E(10, 4) ≈ 5.157$ and $E(100, 10) ≈ 51.893$.

Find $E({10}^{10}, 4000)$. Give your answer rounded to 2 decimal places behind the decimal point.

# --hints--

`rangeFlips()` should return `5000624921.38`.

```js
assert.strictEqual(rangeFlips(), 5000624921.38);
```

# --seed--

## --seed-contents--

```js
function rangeFlips() {

  return true;
}

rangeFlips();
```

# --solutions--

```js
// solution required
```