# Difference between revisions of "Project Euler/53"

### From charlesreid1

(→Solution Explanation) |
m (Replacing charlesreid1.com:3000 with git.charlesreid1.com) |
||

Line 44: | Line 44: | ||

=Solution Code= | =Solution Code= | ||

− | Link: https://charlesreid1.com | + | Link: https://git.charlesreid1.com/cs/euler/src/master/scratch/Round2_050-070/053/CombinatoricSelections.java |

<pre> | <pre> |

## Latest revision as of 03:48, 9 October 2019

# Problem Statement

Problem 53 asks about values of the binomial coefficient, a.k.a. n-choose-k. For some values of n and k, the resulting n-choose-k value can exceed 1 million. This question asks how many binomial coefficients n-choose-k exceed 1 million, for n = 1 to 100.

https://projecteuler.net/problem=53

# Solution Explanation

The solution here creates a matrix of values. We know that if n = k, the binomial coefficient is 1, so we can think of a square matrix, one dimension being n and the other being k, where the values along the diagonal are 1. We now have a discrete set of values on a grid, 100 x 100, and we wish to count how many squares where the n-choose-k value will exceed 1 million.

My clever solution, which enabled me to save a loooot of time that would otherwise be wasted finding factorials, was to use the fact that if n-choose-k is bigger than 1 million, then LOG(n-choose-k) should be bigger than LOG(1 million).

Taking the log of the binomial expression gives:

Now applying log properties simplifies this to:

Further simplifying turns this into a trivial sum:

thus reducing the problem of marking a "square" on our 100 x 100 n vs. k grid reduces to adding numbers that are O(10), instead of multiplying numbers that are O(100000000...)

The final comparison being done is:

If this is true, n-choose-k is less than 1 million; if it is false, n-choose-k is greater than 1 million.

# Solution Code

public class CombinatoricSelections { public static void main(String[] args) { doit(); } /** Nice easy test. This should be bigger than 1 million - should return true. */ public static void test() { System.out.println("isBigger(23,10) = "+isBigger(23,10)); } /** Solve the problem. */ public static void doit() { int MAX = 100; int count = 0; boolean[][] isBig = new boolean[MAX][MAX]; for(int nm1=0; nm1<MAX; nm1++) { for(int rm1=0; rm1<MAX; rm1++) { // determine if n C r > 1M int n = nm1+1; int r = rm1+1; boolean big = isBigger(n,r); isBig[nm1][rm1] = big; if(big) count++; } } System.out.println("Total number of n Choose r values that are greater than 1M: "+count); } /** Boolean: is n choose r bigger than 1 million? */ public static boolean isBigger(int n, int r) { int d = (n-r); double LHS = 0.0; for(int j=0; j<d; j++) { LHS += Math.log(r+d-j) - Math.log(d-j); } double RHS = Math.log(1000000); return LHS > RHS; } }

# Flags

Project Euler
Problem 1
Problem 11
Problem 51
Problem 100
Problem 500
- = in progress
· Template:ProjectEulerFlag · e |