Number of Ways to Split a String

Number of Ways to Split a String

What is this problem about?

The Number of Ways to Split a String problem asks you to split a binary string into 3 non-empty parts that each have the same number of '1's. Count the number of ways modulo 10^9+7. This coding problem uses prefix sum with position enumeration.

Why is this asked in interviews?

Microsoft asks this to test combinatorial counting with prefix sums. The key insight: if total '1's is not divisible by 3, return 0. If total '1's is 0, the answer is C(n-1, 2). Otherwise, find the positions of the 1/3 and 2/3 boundary '1's and count valid split points. The math and string interview pattern is demonstrated.

Algorithmic pattern used

Prefix sum + boundary counting. Count total '1's. If total % 3 != 0, return 0. If total == 0, return C(n-1, 2) (place dividers anywhere between n-1 positions). Otherwise, target = total/3. Find all positions of the target-th '1' (first third boundary) and all positions of the 2*target-th '1' (second third boundary). The number of valid splits = (number of valid positions for first divider) × (number of valid positions for second divider).

Example explanation

s="10101". Total '1's=3. Target=1. First third boundary: 1st '1' at position 0. Positions for first divider: between 0 and 2nd '1' (at position 2) → positions 1,2 (2 choices). Second third boundary: 2nd '1' at 2, 3rd '1' at 4. Positions for second divider: between position 2 and 4 → position 3 (1 choice). Answer = 2*1 = 2.

Common mistakes candidates make

• Not handling total=0 case separately (all zeros → C(n-1,2) ways).
• Off-by-one in divider position counting.
• Using zero-indexed vs one-indexed positions incorrectly.
• Not applying modular arithmetic for the product.

Interview preparation tip

String split-into-equal-property problems use the boundary-counting technique: find where each third of the required property starts/ends, then count valid divider positions (the gaps between boundaries). The key insight is that dividers can be placed in the "zero-filled gaps" between the boundary '1's without changing the count of '1's in each part. Practice "split array into k equal-sum parts" using the same boundary analysis.