<p>The total number of ways of selecting six coins out of some one-rupee coins, 10 fifty-paise coins, and 7 twenty-five paise coins is</p>
Step-by-Step Solution
Key Concept: Since coins of the same denomination are indistinguishable, we only need to count the number of ways to select how many coins of each type (not which specific coins). This reduces to finding non-negative integer solutions with upper bound constraints.
<p><strong>Step 1:</strong> Identify what we're selecting. We need to select 6 coins total from three types: one-rupee (unlimited), fifty-paise (max 10), and twenty-five paise (max 7).</p><p><strong>Step 2:</strong> Let x = number of one-rupee coins, y = number of fifty-paise coins, z = number of twenty-five paise coins.</p><p><strong>Step 3:</strong> We need: x + y + z = 6, where 0 ≤ y ≤ 10 and 0 ≤ z ≤ 7 (and x ≥ 0).</p><p><strong>Step 4:</strong> Since z ≤ 7 and y ≤ 10, but x + y + z = 6, the upper limits are automatically satisfied. We only need non-negative integer solutions to x + y + z = 6.</p><p><strong>Step 5:</strong> By stars and bars: C(6+3-1, 3-1) = C(8,2) = 28.</p><p><strong>Verification:</strong> All solutions have x, y, z ≥ 0 with x + y + z = 6. Since max(y,z) ≤ 6 < 10 and < 7, no upper bound constraints eliminate any solution.</p><p>∴ Answer: <strong>28</strong></p>
Correct Answer: A