<< problem 52 - Permuted multiples Poker hands - problem 54 >>

# Problem 53: Combinatoric selections

There are exactly ten ways of selecting three from five, 12345:

123, 124, 125, 134, 135, 145, 234, 235, 245, and 345

In combinatorics, we use the notation, ^5C_3 = 10.

In general,
^nC_r = dfrac{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: ^{23}C_10 = 1144066.

How many, not necessarily distinct, values of ^nC_r, for 1 <= n <= 100, are greater than one-million?

# Algorithm

The formulas based on factorials (those provided in the problem statement) allow a direct computation of ^nC_r.
However, you might get very big numbers in the numerator and/or denominator - they easily exceed the range of a 32 or 64 bit integer.

There is another formula - the recursive definition:
- ^nC_0 = {^nC_n} = 1 and
- ^nC_k = {^{n-1}C_{k-1}} + {^{n-1}C_k}

As soon as any ^nC_k exceeds maxNumber, I set it to maxNumber + 1. This value still fits into an 64 bit integer (even 2*(maxNumber+1) is no problem).
At the same time, bigNumbers is incremented.

I am allowed to replace ^nC_k by maxNumber + 1 because the true value of ^nC_k doesn't really matter - all we want to know is whether ^nC_k > maxNumber - or not.

When all values are processed, bigNumbers is displayed.

## Note

The program could use less memory: instead of storing all values of combinations[n][k] it is sufficient to keep only
combinations[n-1][] and combinations[n][], thus reducing the memory requirements from maxN^2 to 2*maxN.

# My code

… was written in C++11 and can be compiled with G++, Clang++, Visual C++. You can download it, too.

       #include <vector>
#include <iostream>

int main()
{
// maximum index n and/or k
unsigned int       maxN      = 100;
// what is considered "big" ?
unsigned long long maxNumber = 1000000;
std::cin >> maxN >> maxNumber;

// this will be the displayed result
unsigned int bigNumbers = 0;

// setup a 2D array to hold all values
std::vector<std::vector<unsigned long long>> combinations(maxN + 1);

// C(n,0) = C(n,n) = 1
for (unsigned int n = 0; n <= maxN; n++)
{
combinations[n].resize(n + 1, 0);
combinations[n][0] = combinations[n][n] = 1;
}

// recursive definition:
// C(n,k) = C(n-1, k-1) + C(n-1, k)
for (unsigned int n = 1; n <= maxN; n++)
for (unsigned int k = 1; k < n; k++)
{
auto sum = combinations[n - 1][k - 1] + combinations[n - 1][k];
// clamp numbers to avoid exceeding 64 bits
if (sum > maxNumber)
{
sum = maxNumber + 1;
// we found one more big number
bigNumbers++;
}

// store result
combinations[n][k] = sum;
}

std::cout << bigNumbers << std::endl;
return 0;
}


This solution contains 7 empty lines, 10 comments and 2 preprocessor commands.

# Interactive test

You can submit your own input to my program and it will be instantly processed at my server:

Input data (separated by spaces or newlines):

This is equivalent to
echo "23 1000000" | ./53

Output:

(this interactive test is still under development, computations will be aborted after one second)

# Benchmark

The correct solution to the original Project Euler problem was found in less than 0.01 seconds on a Intel® Core™ i7-2600K CPU @ 3.40GHz.
(compiled for x86_64 / Linux, GCC flags: -O3 -march=native -fno-exceptions -fno-rtti -std=c++11 -DORIGINAL)

See here for a comparison of all solutions.

Note: interactive tests run on a weaker (=slower) computer. Some interactive tests are compiled without -DORIGINAL.

# Changelog

February 27, 2017 submitted solution

# Hackerrank

My code solved 5 out of 5 test cases (score: 100%)

# Difficulty

Project Euler ranks this problem at 5% (out of 100%).

Hackerrank describes this problem as easy.

Note:
Hackerrank has strict execution time limits (typically 2 seconds for C++ code) and often a much wider input range than the original problem.
In my opinion, Hackerrank's modified problems are usually a lot harder to solve. As a rule thumb: brute-force is never an option.

projecteuler.net/thread=53 - the best forum on the subject (note: you have to submit the correct solution first)

Code in various languages:

Python: www.mathblog.dk/project-euler-53-cnr-exceed-one-million/ (written by Kristian Edlund)
Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p053.java (written by Nayuki)
Mathematica: github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p053.mathematica (written by Nayuki)
Go: github.com/frrad/project-euler/blob/master/golang/Problem053.go (written by Frederick Robinson)
Javascript: github.com/dsernst/ProjectEuler/blob/master/53 Combinatoric selections.js (written by David Ernst)
Scala: github.com/samskivert/euler-scala/blob/master/Euler053.scala (written by Michael Bayne)

# Heatmap

green problems solve the original Project Euler problem and have a perfect score of 100% at Hackerrank, too.
yellow problems score less than 100% at Hackerrank (but still solve the original problem).
gray problems are already solved but I haven't published my solution yet.
blue problems are already solved and there wasn't a Hackerrank version of it (at the time I solved it) or I didn't care about it because it differed too much.

Please click on a problem's number to open my solution to that problem:

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The 133 solved problems had an average difficulty of 16.9% at Project Euler and I scored 11,174 points (out of 12300) at Hackerrank's Project Euler+.
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