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Problem 76: Counting summations

curriculum/challenges/english/blocks/project-euler-problems-1-to-100/5900f3b81000cf542c50fecb.md

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--description--

It is possible to write five as a sum in exactly six different ways:

<div style='margin-left: 4em;'> 4 + 1

3 + 2

3 + 1 + 1

2 + 2 + 1

2 + 1 + 1 + 1

1 + 1 + 1 + 1 + 1

</div>

How many different ways can n be written as a sum of at least two positive integers?

--hints--

countingSummations(5) should return a number.

js
assert(typeof countingSummations(5) === 'number');

countingSummations(5) should return 6.

js
assert.strictEqual(countingSummations(5), 6);

countingSummations(20) should return 626.

js
assert.strictEqual(countingSummations(20), 626);

countingSummations(50) should return 204225.

js
assert.strictEqual(countingSummations(50), 204225);

countingSummations(100) should return 190569291.

js
assert.strictEqual(countingSummations(100), 190569291);

--seed--

--seed-contents--

js
function countingSummations(n) {

  return true;
}

countingSummations(5);

--solutions--

js
function countingSummations(n) {
  const combinations = new Array(n + 1).fill(0);
  combinations[0] = 1;

  for (let i = 1; i < n; i++) {
    for (let j = i; j < n + 1; j++) {
      combinations[j] += combinations[j - i];
    }
  }
  return combinations[n];
}