The Number of Distinct Binary Strings After Applying Operations problem gives you a binary string s and an integer k. In one operation, you can choose any k consecutive characters and flip all of them. Determine how many distinct binary strings can be produced after any number of operations. This coding problem requires recognizing a mathematical pattern about independent operation positions.
Microsoft asks this to test the ability to derive a closed-form mathematical answer from pattern recognition. The key insight: each starting position where a k-length window begins is an independent "degree of freedom." The math and string interview pattern is demonstrated through combinatorial reasoning.
Mathematical observation. There are n - k + 1 valid starting positions for the k-length window. Each starting position can be flipped or not independently (the flips from different positions can interact, but the total degrees of freedom = n-k+1). Answer: 2^(n-k+1) modulo 10^9+7.
s="0000", k=2. n=4. Valid positions: 0,1,2. Degrees of freedom = 3. Answer = 2^3 = 8 distinct strings. Verify: we can produce 8 different strings from flipping choices at positions 0,1,2 independently.
s="01", k=2. n=2. Positions: 0. Degrees=1. Answer = 2^1 = 2 (original and fully-flipped).
n - k + 1, not n - k.When an operation on a string affects a fixed-length window, count the number of independent windows — each is a binary choice (flip or not). The answer is 2^(number of windows). This pattern of "count independent degrees of freedom → 2^count" appears in XOR flip problems, bit toggle problems, and many combinatorial string problems. Practice deriving such formulas from small examples (n=4, k=2; n=5, k=3) before generalizing.