<p>In an experiment, \(n\) six-faced normal dice are thrown. Find the number of sets of observations which are indistinguishable among themselves.</p>
Step-by-Step Solution
Key Concept: This is a 'stars and bars' problem: we need to count non-negative integer solutions to x₁ + x₂ + ... + x₆ = n, where xᵢ represents how many dice show face i. The number of indistinguishable sets corresponds to multisets of size n from 6 distinct outcomes.
<p><strong>Step 1:</strong> Recognize what 'indistinguishable sets' means. Two observations are indistinguishable if they have the same frequency distribution across faces. For example, {1,1,2,3} is indistinguishable from {2,1,3,1} because both have two 1's, one 2, and one 3.</p><p><strong>Step 2:</strong> Let xᵢ = frequency of face i appearing (where i = 1,2,3,4,5,6). We need: x₁ + x₂ + x₃ + x₄ + x₅ + x₆ = n, where each xᵢ ≥ 0.</p><p><strong>Step 3:</strong> This is a 'distributing n identical objects into 6 distinct bins' problem. By stars and bars formula, the number of non-negative integer solutions is:</p><p>$$\binom{n + 6 - 1}{6 - 1} = \binom{n + 5}{5}$$</p><p><strong>Verification:</strong> For n=1: \(\binom{6}{5} = 6\) (one die can show any of 6 faces) ✓</p><p>∴ Answer: \(\binom{n+5}{5}\)</p>
Correct Answer: \(\binom{n+5}{5}\)