Add Pascal's triangle.

Author: trekhleb

src/algorithms/math/pascal-triangle/README.md

@@ -1,7 +1,7 @@
 # Pascal's Triangle
 
 In mathematics, **Pascal's triangle** is a triangular array of 
-the binomial coefficients.
+the [binomial coefficients](https://en.wikipedia.org/wiki/Binomial_coefficient).
 
 The rows of Pascal's triangle are conventionally enumerated 
 starting with row `n = 0` at the top (the `0th` row). The 
@@ -34,6 +34,31 @@ paragraph may be written as follows:
 for any non-negative integer `n` and any 
 integer `k` between `0` and `n`, inclusive.
 
+![Binomial Coefficient](https://wikimedia.org/api/rest_v1/media/math/render/svg/a2457a7ef3c77831e34e06a1fe17a80b84a03181)
+
+## Calculating triangle entries in O(n) time
+
+We know that `i`-th entry in a line number `lineNumber` is 
+Binomial Coefficient `C(lineNumber, i)` and all lines start 
+with value `1`. The idea is to 
+calculate `C(lineNumber, i)` using `C(lineNumber, i-1)`. It 
+can be calculated in `O(1)` time using the following:
+
+```
+C(lineNumber, i) = lineNumber! / ((lineNumber - i)! * i!)
+C(lineNumber, i - 1) = lineNumber! / ((lineNumber - i + 1)! * (i - 1)!)
+```
+
+We can derive following expression from above two expressions:
+
+```
+C(lineNumber, i) = C(lineNumber, i - 1) * (lineNumber - i + 1) / i
+```
+
+So `C(lineNumber, i)` can be calculated 
+from `C(lineNumber, i - 1)` in `O(1)` time.
+
 ## References
 
 - [Wikipedia](https://en.wikipedia.org/wiki/Pascal%27s_triangle)
+- [GeeksForGeeks](https://www.geeksforgeeks.org/pascal-triangle/)

src/algorithms/math/pascal-triangle/__test__/pascalTriangle.test.js

@@ -0,0 +1,14 @@
+import pascalTriangle from '../pascalTriangle';
+
+describe('pascalTriangle', () => {
+ it('should calculate Pascal Triangle coefficients for specific line number', () => {
+ expect(pascalTriangle(0)).toEqual([1]);
+ expect(pascalTriangle(1)).toEqual([1, 1]);
+ expect(pascalTriangle(2)).toEqual([1, 2, 1]);
+ expect(pascalTriangle(3)).toEqual([1, 3, 3, 1]);
+ expect(pascalTriangle(4)).toEqual([1, 4, 6, 4, 1]);
+ expect(pascalTriangle(5)).toEqual([1, 5, 10, 10, 5, 1]);
+ expect(pascalTriangle(6)).toEqual([1, 6, 15, 20, 15, 6, 1]);
+ expect(pascalTriangle(7)).toEqual([1, 7, 21, 35, 35, 21, 7, 1]);
+ });
+});

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

@@ -0,0 +1,16 @@
+/**
+ * @param {number} lineNumber - zero based.
+ * @return {number[]}
+ */
+export default function pascalTriangle(lineNumber) {
+ const currentLine = [1];
+
+ const currentLineSize = lineNumber + 1;
+
+ for (let numIndex = 1; numIndex < currentLineSize; numIndex += 1) {
+ // See explanation of this formula in README.
+ currentLine[numIndex] = currentLine[numIndex - 1] * (lineNumber - numIndex + 1) / numIndex;
+ }
+
+ return currentLine;
+}

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

@@ -1,5 +1,5 @@
 /**
- * @param {number} lineNumber
+ * @param {number} lineNumber - zero based.
  * @return {number[]}
  */
 export default function pascalTriangleRecursive(lineNumber) {

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

@@ -1,5 +1,6 @@
 import combineWithoutRepetitions from '../combineWithoutRepetitions';
 import factorial from '../../../math/factorial/factorial';
+import pascalTriangle from '../../../math/pascal-triangle/pascalTriangle';
 
 describe('combineWithoutRepetitions', () => {
  it('should combine string without repetitions', () => {
@@ -56,5 +57,8 @@ describe('combineWithoutRepetitions', () => {
  const expectedNumberOfCombinations = factorial(n) / (factorial(r) * factorial(n - r));
 
  expect(combinations.length).toBe(expectedNumberOfCombinations);
+
+ // This one is just to see one of the way of Pascal's triangle application.
+ expect(combinations.length).toBe(pascalTriangle(n)[r]);
  });
 });