#include

using namespace std;

$//$ Function to calculate permutations $($N choose X$)$

long long Permutations$($long long N$,$ long long X$)\{$

$$ long long result $=1$;

$$ for $($long long i $=0$; i $<$ X; $++$i$)\{$

$$ result $*=($N $-$ i$)$;

$\}$

$$ return result;

$\}$

$//$ Function to calculate combinations $($N choose X$)$

long long Combinations$($long long N$,$ long long X$)\{$

$$ if $($X $>$ N $-$ X$)$ X $=$ N $-$ X;

$$ long long result $=1$;

$$ for $($long long i $=0$; i $<$ X; $++$i$)\{$

$$ result $*=($N $-$ i$)$;

$$ result $/=($i $+1)$;

$\}$

$$ return result;

$\}$

int main$()\{$

$//1.$ Hospital emergency room problem

$$ long long doctors $=12$;

$$ long long nurses $=36$;

$$ long long doctorsInBatch $=4$;

$$ long long nursesInBatch $=12$;

$$

$//1$a$.$ Ways to select doctors for the first set of doses

$$ long long waysToSelectDoctors $=$ Combinations$($doctors$,$ doctorsInBatch$)$;

$$ cout $<<"1$a$.$ Ways to select doctors: $"<<$ waysToSelectDoctors $<<$ endl;

$//1$b$.$ Ways to select nurses for the first set of doses

$$ long long waysToSelectNurses $=$ Combinations$($nurses$,$ nursesInBatch$)$;

$$ cout $<<"1$b$.$ Ways to select nurses: $"<<$ waysToSelectNurses $<<$ endl;

$//1$c$.$ Ways to administer the first dose

$$ long long totalPeople $=$ doctors $+$ nurses;

$$ long long waysToAdministerFirstDose $=$ Permutations$($totalPeople$,$ doctorsInBatch $+$ nursesInBatch$)$;

$$ cout $<<"1$c$.$ Ways to administer the first dose: $"<<$ waysToAdministerFirstDose $<<$ endl;

$//1$d$.$ Ways to administer the second dose

$$ long long remainingPeople $=$ totalPeople $-($doctorsInBatch $+$ nursesInBatch$)$;

$$ long long waysToAdministerSecondDose $=$ Permutations$($remainingPeople$,$ doctorsInBatch $+$ nursesInBatch$)$;

$$ cout $<<"1$d$.$ Ways to administer the second dose: $"<<$ waysToAdministerSecondDose $<<$ endl;

$//1$e$.$ Total ways to administer all doses

$$ long long totalWays $=$ waysToSelectDoctors $*$ waysToSelectNurses $*$ waysToAdministerFirstDose $*$ waysToAdministerSecondDose;

$$ cout $<<"1$e$.$ Total ways to administer all doses: $"<<$ totalWays $<<$ endl;

$$ cout $<<$ "Formula: waysToSelectDoctors $*$ waysToSelectNurses $*$ waysToAdministerFirstDose $*$ waysToAdministerSecondDose

$"<<$ endl;

$//2.$ Bonuses distribution problem

$$ long long employees $=23$;

$$ long long bonuses $=4$;

$//2$a$.$ Ways to distribute different bonuses

$$ long long waysToDistributeBonuses $=$ Permutations$($employees$,$ bonuses$)$;

$$ cout $<<$ "The number of ways to distribute $4$ bonuses to $23$ people if different is: $"<<$ waysToDistributeBonuses $<<$ endl;

$//2$b$.$ Ways to distribute identical bonuses

$$ long long waysToDistributeIdenticalBonuses $=$ Combinations$($employees$,$ bonuses$)$;

$$ cout $<<"$If the bonuses are the same, there are: $"<<$ waysToDistributeIdenticalBonuses $<<$ endl;

$//$ Test the functions

$$ long long permTest $=$ Permutations$(35,12)$;

$$ cout $<<"$

Permutations of $35$ things taken $12$ at a time: $"<<$ permTest $<<$ endl;

$$ long long combTest $=$ Combinations$(23,11)$;

$$ cout $<<$ "Combinations of $23$ things taken $11$ at a time: $"<<$ combTest $<<$ endl;

$$ return $0$;

$\}$