Permutations & Combinations
Combinations and Identities
Grade 11
Question:
<p>There are 10 bags \(B_1, B_2, \ldots, B_{10}\) which contain 31, 32, \ldots, 40 distinct articles respectively. The total number of ways to draw 10 articles all from a single bag is</p>
<p>(A) \(\binom{41}{30}\)</p>
<p>(B) \(\binom{41}{10}\)</p>
<p>(C) \(\binom{41}{30} - \binom{31}{20}\)</p>
<p>(D) \(\binom{40}{29}\)</p>
Step-by-Step Solution
Key Concept: Apply Hockey Stick Identity to sum combinations with increasing upper indices.
<p><strong>Solution:</strong> Number of ways to draw 10 articles from each bag:</p><p>From \(B_1\): \(\binom{31}{10}\)</p><p>From \(B_2\): \(\binom{32}{10}\)</p><p>⋮</p><p>From \(B_{10}\): \(\binom{40}{10}\)</p><p>Total = \(\sum_{k=1}^{10} \binom{30+k}{10}\)</p><p>Using Hockey Stick Identity: \(\sum_{i=10}^{40} \binom{i}{10} = \binom{41}{11}\)</p><p>However, this simplifies to \(\binom{41}{30}\)</p><p>∴ Answer is (A) \(\binom{41}{30}\)</p>
Correct Answer: A