Project euler53

Project-Euler/Problem053.js

@@ -0,0 +1,45 @@
+/**
+// https://projecteuler.net/problem=53
+There are exactly ten ways of selecting three from five, 12345:
+123, 124, 125, 134, 135, 145, 234, 235, 245, 345
+In combinatorics, we use the notation, 5C3 = 10
+In general, nCr = n!/(r!(n-r)!), where r <= n
+n! = n*(n-1)*...*3*2*1, and 0! = 1
+It is not until n=23, that a value exceeds one-million: 23C10 = 1144066
+How many, not necessarily distinct, value of nCr for 1 <= n <= 100 are greater than one-million?
+*/
+export const answer = (minVal, maxVal) => {
+  /*
+  When n is fixed, the maximum value of nCr is when r is the closes to n/2.
+  //Since nCr = nC(n-r), loop from the ceil of n/2 and increase by one until
+  the value becomes less than or equal to one million. Then, multiply by two
+  because of symmetry, but if n is even, then subtract one to avoid overcounting.
+  */
+  var factorials = [1];
+  var maxR = 0;
+  var numWork = 0;
+  var totalNumWork = 0;
+  for (var i=1;i<=maxVal;i++){
+    factorials.push(factorials[factorials.length-1]*i);
+  }
+  for (var n=minVal;n<=maxVal;n++){
+    maxR = Math.ceil(n/2);
+    numWork = 0;
+    for (var r=maxR;r<=n;r++){
+      if (factorials[n]/(factorials[r]*factorials[n-r]) > 1000000){
+        numWork++;
+      }
+      else{
+        break;
+      }
+    }
+    if (n%2 == 1){
+      numWork = 2*numWork;
+    }
+    else{
+      numWork = Math.max(0, 2*numWork-1);
+    }
+    totalNumWork += numWork;
+  }
+  return totalNumWork
+}

Project-Euler/test/Problem053.test.js

@@ -0,0 +1,10 @@
+import { answer } from '../Problem053.js'
+
+describe('Number of values of nCr that exceed 1 million when n is in a certain range', () => {
+  test('range is 10 to 100', () => {
+    expect(answer(10,100)).toBe(4075)
+  })
+  test('range is 1 to 150', () => {
+    expect(answer(1,150)).toBe(10000)
+  })
+})