There are only 2 valid patterns: ‘101’ and ‘010’. Think about how we can construct these 2 patterns from smaller patterns.

Count the number of subsequences of the form ‘01’ or ‘10’ first. Let n01[i] be the number of ‘01’ subsequences that exist in the prefix of s up to the ith building. How can you compute n01[i]?

Let n0[i] and n1[i] be the number of ‘0’s and ‘1’s that exists in the prefix of s up to i respectively. Then n01[i] = n01[i – 1] if s[i] == ‘0’, otherwise n01[i] = n01[i – 1] + n0[i – 1].

The same logic applies to building the n10 array and subsequently the n101 and n010 arrays for the number of ‘101’ and ‘010‘ subsequences.