<p>The number of ways in which 7 identical rings can be put on 5 fingers of a hand is ______, if each finger has at least one ring.</p>
Step-by-Step Solution
Key Concept: This is a stars and bars problem with a constraint: we need to distribute 7 identical objects into 5 distinct groups with each group getting at least one. Transform it by giving 1 ring to each finger first, then distribute the remaining 2 rings freely.
<p><strong>Step 1:</strong> Recognize this as a distribution problem where 7 identical rings must be placed on 5 distinct fingers, with each finger getting at least one ring.</p><p><strong>Step 2:</strong> Since each of the 5 fingers must have at least one ring, give 1 ring to each finger. This uses 5 rings and leaves 7 - 5 = 2 rings to distribute freely.</p><p><strong>Step 3:</strong> Now distribute the remaining 2 identical rings among 5 distinct fingers with no restrictions. Use stars and bars formula: C(n+k-1, k-1) where n = 2 remaining rings and k = 5 fingers.</p><p><strong>Step 4:</strong> Number of ways = C(2+5-1, 5-1) = C(6, 4) = C(6, 2) = (6 × 5)/(2 × 1) = 15</p><p>∴ Answer: <strong>15</strong></p>
Correct Answer: 15