<p>If <i>N</i> denotes the number of ways of selecting <i>r</i> objects out of <i>n</i> distinct objects (<i>r</i> ≥ <i>n</i>) with unlimited repetition but with each object included at least once in selection, then <i>N</i> is equal to</p>
Step-by-Step Solution
Key Concept: When r ≥ n objects are selected from n distinct objects with repetition allowed and each object must appear at least once, first guarantee one copy of each object (uses n selections), then distribute the remaining (r-n) selections freely among n objects.
<p><strong>Step 1:</strong> Recognize that r ≥ n and each of n distinct objects must appear at least once in the selection.</p><p><strong>Step 2:</strong> Allocate one copy of each object first. This uses n selections and satisfies the 'at least once' constraint.</p><p><strong>Step 3:</strong> We now have (r - n) remaining selections to distribute freely among the n objects with no restrictions.</p><p><strong>Step 4:</strong> Apply Stars and Bars formula for distributing (r-n) identical items into n distinct groups: C(n + (r-n) - 1, r-n) = C(r-1, r-n) = C(r-1, n-1)</p><p><strong>Step 5:</strong> Alternatively, this equals <sup>n+r-1-n</sup>C<sub>r-n</sub> = <sup>r-1</sup>C<sub>n-1</sub></p><p>∴ Answer: C (N = <sup>r-1</sup>C<sub>n-1</sub>)</p>
Correct Answer: C