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Count even length binary sequences with same sum of first and second half bits in C++
We are given several bits n as input for a binary sequence. The goal here is to find the binary sequence of length 2n such that the sum of its first and second half bits is equal. First n bits and next n bits have the same sum.
We have a binary sequence so the only choice to put digits at any place is 0 and 1. For n bits at first and second half, no. of possible combinations are −
n bits with all zeros (0 1’s) nC0= 1
n bits with 1 1’s nC1
n bits with 2 1’s nC2
.
.
n bits with n 1’s nCn
For 2n bits
First half with 0 1’s and second half with 0 1’s nC0 X nC0
First half with 1 1’s and second half with 1 1’s nC1 X nC1
First half with 2 1’s and second half with 2 1’s nC2 X nC2
..............
First half with n 1’s and second half with n 1’s nCn X nCn
Total such combinations= nC0*nC0 + nC1*nC1+.......+nCn*nCn
=(nC0)2+(nC1)2+...........+(nCn)2
Input
n=1
Output
Sequences with same sum of first and second half bits: 2
Explanation − Possible 2*1=2 bit sequences 00,01,10,11 Out of these four 01 and 10 have sum=1
Input
n=2
Output
Sequences with same sum of first and second half bits: 6
Explanation − Possible 2*2 = 4-bit sequences 0000,0001,0010,0011,0100,0101,0110,0111,1000,1001,1010,1011,1100,1101,1110,1111
Out of these sequences with sum of first 2 and last 2 bits is same −
0000,0101,0110,1001,1010,1111, total=6
Approach used in the below program is as follows
Integer ‘bits’ stores the number.
Function findSeq(int n) takes n as input and returns the count of sequences with above sum of first and second half 2n bits equal.
Variable nCi is used to store the initial value =1 as nC0 is 1.
Initialize ans=0, which will count such sequences as sum of nCi*nCi.
Starting from i=0 to n add nCi*nCi to the ans, calculate each nCi as above formula.
After the end of for loop return the result present in ‘ans’ as count.
Example
#include<iostream>
using namespace std;
// Returns the count of even length sequences
int findSeq(int n){
int nCi=1; //nC0=1
int ans = 1;
for (int i = 1; i<=n ; i++){
//nCi=(nCi-1)*(nCi/nCi-1)
// nCi/nC(i-1) = (n+1-i)/i;
nCi = (nCi * (n+1-i))/i;
ans += nCi*nCi;
}
return ans;
}
int main(){
int bits = 2;
cout << "Count of binary sequences such that sum of first and second half bits is
same: "<<findSeq(bits);
return 0;
}Output
Count of binary sequences such that sum of first and second half bits is same: 6
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