<< 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?

# 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 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.

# 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 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:*

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 |

26 | 27 | 28 | 29 | 30 | 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 | 41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 | 50 |

51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 | 60 | 61 | 62 | 63 | 64 | 65 | 66 | 67 | 68 | 69 | 70 | 71 | 72 | 73 | 74 | 75 |

76 | 77 | 78 | 79 | 80 | 81 | 82 | 83 | 84 | 85 | 86 | 87 | 88 | 89 | 90 | 91 | 92 | 93 | 94 | 95 | 96 | 97 | 98 | 99 | 100 |

101 | 102 | 103 | 104 | 105 | 106 | 107 | 108 | 109 | 110 | 111 | 112 | 113 | 114 | 115 | 116 | 117 | 118 | 119 | 120 | 121 | 122 | 123 | 124 | 125 |

126 | 127 | 128 | 129 | 130 | 131 | 132 | 133 | 134 | 135 | 136 | 137 | 138 | 139 | 140 | 141 | 142 | 143 | 144 | 145 | 146 | 147 | 148 | 149 | 150 |

151 | 152 | 153 | 154 | 155 | 156 | 157 | 158 | 159 | 160 | 161 | 162 | 163 | 164 | 165 | 166 | 167 | 168 | 169 | 170 | 171 | 172 | 173 | 174 | 175 |

176 | 177 | 178 | 179 | 180 | 181 | 182 | 183 | 184 | 185 | 186 | 187 | 188 | 189 | 190 | 191 | 192 | 193 | 194 | 195 | 196 | 197 | 198 | 199 | 200 |

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