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

# Problem 53: Combinatoric selections

(see projecteuler.net/problem=53)

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?

# My 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:

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

April 20, 2017 added comments

# Hackerrank

see https://www.hackerrank.com/contests/projecteuler/challenges/euler053

My code solves **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 rarely an option.

# Links

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)

Haskell: github.com/nayuki/Project-Euler-solutions/blob/master/haskell/p053.hs (written by Nayuki)

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 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.

red problems are solved but exceed the time limit of one minute or the memory limit of 256 MByte.

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

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I scored 12,983 points (out of 15100 possible points, top rank was 17 out of ≈60000 in August 2017) at Hackerrank's Project Euler+.

Look at my progress and performance pages to get more details.

My username at Project Euler is

**stephanbrumme**while it's stbrumme at Hackerrank.

# Copyright

I hope you enjoy my code and learn something - or give me feedback how I can improve my solutions.

All of my solutions can be used for any purpose and I am in no way liable for any damages caused.

You can even remove my name and claim it's yours. But then you shall burn in hell.

The problems and most of the problems' images were created by Project Euler. Thanks for all their endless effort.

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