Introduction to ACM [inclusion exclusion theorem]

It may be fascinating to see the picture , Let's look directly at the example .
Example 1 :

Very classic content exclusion , We can use binary enumeration to enumerate all collections .
For the sign of each set , Is the number of sets in the set . for example :

#include<bits/stdc++.h>
using namespace std;
typedef long long int LL;
LL a[25],n,m,sum;
LL solve(LL n)
{
    
    LL sum=0;
    for(int i=1;i<(1<<m);i++)// Attention from 1 Start 
    {
    
        LL temp=n,cnt=-1;
        for(int j=0;j<m;j++)
        {
    
            if(i>>j&1)
            {
    
                temp/=a[j],cnt=cnt*(-1);
            }
        }
        sum=sum+cnt*temp;
    }
    return sum;
}
int main(void)
{
    
    cin>>n>>m;
    for(int i=0;i<m;i++) cin>>a[i];
    cout<<solve(n);
    return 0;
}

Example 2 ：

#include<bits/stdc++.h>
using namespace std;
typedef long long int LL;
const int N=25;
LL l,r,n,a[N];
LL solve(LL x)
{
    
	LL sum=0;
	for(int i=1;i<(1<<n);i++)
	{
    
		LL temp=x,cnt=-1;
		for(int j=0;j<n;j++)
		{
    
			if((i>>j)&1==1) temp/=a[j],cnt=cnt*(-1);
		}
		sum=sum+cnt*temp;
	}
	return x-sum;
}
int main(void)
{
    
	cin>>l>>r>>n;
	for(int i=0;i<n;i++) cin>>a[i];
	cout<<solve(r)-solve(l-1);
	return 0;
}