Number of non-negative integer solutions without restriction: C(10 + 5 - 1, 5 - 1) = C(14,4) = <<comb(14,4) = 1001>>1001

Wed, 06 May 2026 10:04:48 +0000

**What Is the Number of Non-Negative Integer Solutions Without Restriction? A Deep Dive** Ever wondered how many ways there are to distribute 5 identical items among 10 distinct bins—counting every possible arrangement, even ones that seem unusual? This question leads to a foundational calculation in combinatorics: the number of non-negative integer solutions to equations of the form x₁ + x₂ + ... + x₅ = n, where variables represent counts and n is the total amount. When parameters are 10 and 5 (plus an adjustment for unrestricted sums), the solution becomes a classic: C(10 + 5 - 1, 5 - 1), or comb(14,4), which equals 1,001. This result—1,001 distinct arrangements—might seem abstract, yet it shapes real patterns in data, algorithms, and even betting models. Understanding it reveals hidden structures behind seemingly complex systems. **Why C(14,4) = 1001 Is Gaining Relevant Attention in the US**