# 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.

# 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:

$\log \left( \dfrac{n!}{k! (n-k)!} \right) = \log \left[ \dfrac{n \times (n-1) \times \dots \times 1 }{ ( k \times (k-1) \times \dots \times 1 )( (n-k) \times (n-k-1) \times \dots \times 1 )} \right]$

Now applying log properties simplifies this to:

$\log \left( \dfrac{n!}{k! (n-k)!} \right) = \log \left[ n \times (n-1) \times \dots \times 1 \right] - \log \left[ k \times (k-1) \times \dots \times 1 \right] - \log \left[ (n-k) \times (n-k-1) \times \dots \times 1 \right]$

Further simplifying turns this into a trivial sum:

$\log \left( \dfrac{n!}{k! (n-k)!} \right) = \sum_{p=1}^{n} \log p - \sum_{q=1}^{n-k} \log q - \sum_{r=1}^{k} \log r$

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:

$\sum_{p=1}^{n} \log p - \sum_{q=1}^{n-k} \log q - \sum_{r=1}^{k} \log r < \log(10000)$

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;
}
}