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Commit b878390

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Add Pascal's triangle.
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‎src/algorithms/math/pascal-triangle/README.md‎

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# Pascal's Triangle
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In mathematics, **Pascal's triangle** is a triangular array of
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the binomial coefficients.
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the [binomial coefficients](https://en.wikipedia.org/wiki/Binomial_coefficient).
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The rows of Pascal's triangle are conventionally enumerated
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starting with row `n = 0` at the top (the `0th` row). The
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for any non-negative integer `n` and any
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integer `k` between `0` and `n`, inclusive.
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![Binomial Coefficient](https://wikimedia.org/api/rest_v1/media/math/render/svg/a2457a7ef3c77831e34e06a1fe17a80b84a03181)
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## Calculating triangle entries in O(n) time
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We know that `i`-th entry in a line number `lineNumber` is
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Binomial Coefficient `C(lineNumber, i)` and all lines start
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with value `1`. The idea is to
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calculate `C(lineNumber, i)` using `C(lineNumber, i-1)`. It
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can be calculated in `O(1)` time using the following:
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```
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C(lineNumber, i) = lineNumber! / ((lineNumber - i)! * i!)
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C(lineNumber, i - 1) = lineNumber! / ((lineNumber - i + 1)! * (i - 1)!)
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```
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We can derive following expression from above two expressions:
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```
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C(lineNumber, i) = C(lineNumber, i - 1) * (lineNumber - i + 1) / i
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```
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So `C(lineNumber, i)` can be calculated
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from `C(lineNumber, i - 1)` in `O(1)` time.
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## References
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- [Wikipedia](https://en.wikipedia.org/wiki/Pascal%27s_triangle)
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- [GeeksForGeeks](https://www.geeksforgeeks.org/pascal-triangle/)
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import pascalTriangle from '../pascalTriangle';
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describe('pascalTriangle', () => {
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it('should calculate Pascal Triangle coefficients for specific line number', () => {
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expect(pascalTriangle(0)).toEqual([1]);
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expect(pascalTriangle(1)).toEqual([1, 1]);
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expect(pascalTriangle(2)).toEqual([1, 2, 1]);
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expect(pascalTriangle(3)).toEqual([1, 3, 3, 1]);
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expect(pascalTriangle(4)).toEqual([1, 4, 6, 4, 1]);
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expect(pascalTriangle(5)).toEqual([1, 5, 10, 10, 5, 1]);
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expect(pascalTriangle(6)).toEqual([1, 6, 15, 20, 15, 6, 1]);
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expect(pascalTriangle(7)).toEqual([1, 7, 21, 35, 35, 21, 7, 1]);
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});
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});
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/**
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* @param {number} lineNumber - zero based.
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* @return {number[]}
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*/
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export default function pascalTriangle(lineNumber) {
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const currentLine = [1];
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const currentLineSize = lineNumber + 1;
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for (let numIndex = 1; numIndex < currentLineSize; numIndex += 1) {
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// See explanation of this formula in README.
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currentLine[numIndex] = currentLine[numIndex - 1] * (lineNumber - numIndex + 1) / numIndex;
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}
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return currentLine;
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}

‎src/algorithms/math/pascal-triangle/pascalTriangleRecursive.js‎

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/**
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* @param {number} lineNumber
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* @param {number} lineNumber - zero based.
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* @return {number[]}
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*/
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export default function pascalTriangleRecursive(lineNumber) {

‎src/algorithms/sets/combinations/__test__/combineWithoutRepetitions.test.js‎

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import combineWithoutRepetitions from '../combineWithoutRepetitions';
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import factorial from '../../../math/factorial/factorial';
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import pascalTriangle from '../../../math/pascal-triangle/pascalTriangle';
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describe('combineWithoutRepetitions', () => {
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it('should combine string without repetitions', () => {
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const expectedNumberOfCombinations = factorial(n) / (factorial(r) * factorial(n - r));
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expect(combinations.length).toBe(expectedNumberOfCombinations);
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// This one is just to see one of the way of Pascal's triangle application.
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expect(combinations.length).toBe(pascalTriangle(n)[r]);
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});
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});

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