How many 6-digit positive integers consist only of the digits 1 and 2, and contain at least two consecutive 1s?

Mon, 04 May 2026 04:25:11 +0000

**How many 6-digit positive integers consist only of the digits 1 and 2, and contain at least two consecutive 1s?**
Curious readers often ask: how many 6-digit positive integers made up only of digits 1 and 2 contain at least two consecutive 1s? This inquiry reflects growing interest in patterns embedded in combinatorial data, particularly as more people explore structured number sets for both practical and recreational purposes. Understanding how many such 6-digit numbers exist isn’t just an academic exercise—it reveals insight into binary-digit combinations and their unique properties. How many 6-digit positive integers consist only of the digits 1 and 2, and contain at least two consecutive 1s? The answer lies in calculating total valid combinations and subtracting those that don’t meet the continuity requirement. There are 2⁶ = 64 total 6-digit numbers using only 1 and 2. But only a subset contains at least one pair of consecutive 1s—a detail that reveals a fascinating pattern in binary sequences. **Why This Question Is Gaining Attention in the US**
In recent years, interest in combinatorics and number patterns has surged, driven by popularity in education, puzzles, and coding practices. Americans exploring self-education, data curiosity, or even personal finance often encounter sequences and digit rules. The question taps into this broader curiosity—about scarcity, frequency, and structured randomness—making it relevant beyond niche math enthusiasts. **How Many Valid Combinations Exist?**
To determine how many 6-digit numbers composed only of 1s and 2s have at least two consecutive 1s, first calculate the total combinations: 2⁶ = 64. Then compute the counter—the count of sequences that *do not* contain “11” as a consecutive pair. Using methods like recursive counting or dynamic programming, experts find there are 21 such sequences without consecutive 1s. Subtracting gives: 64 – 21 = 43.
So, 43 six-digit numbers made exclusively of 1s and 2s *do* contain at least two consecutive 1s.