<p>Find the number of ways of selecting 10 objects from 42 objects of 21 objects are identical and remaining objects are distinct.</p>
Step-by-Step Solution
Key Concept: Partition the selection problem by how many identical objects are chosen (0 to 10), then for each case count ways to select remaining objects from the 21 distinct ones. This gives a sum that telescopes to 2^20.
<p><strong>Step 1:</strong> Partition by number of identical objects selected.</p><p>Let i = number of identical objects chosen (where 0 ≤ i ≤ 10).</p><p>Then we must choose (10 - i) objects from the 21 distinct objects.</p><p><strong>Step 2:</strong> For each value of i, there is exactly 1 way to choose i identical objects and C(21, 10-i) ways to choose (10-i) distinct objects.</p><p><strong>Step 3:</strong> Total number of ways = Σ(i=0 to 10) C(21, 10-i)</p><p>Substituting j = 10-i, this becomes: Σ(j=0 to 10) C(21, j)</p><p><strong>Step 4:</strong> By the hockey-stick identity and symmetry property:</p><p>Σ(j=0 to 10) C(21, j) = Σ(j=0 to 10) C(21, 21-j) = Σ(k=11 to 21) C(21, k)</p><p>Since Σ(j=0 to 21) C(21, j) = 2^21 and by symmetry Σ(j=0 to 10) C(21, j) = Σ(j=11 to 21) C(21, j), we have:</p><p>2 × Σ(j=0 to 10) C(21, j) = 2^21 - C(21, 10) [accounting for middle term correction]</p><p>Actually: Σ(j=0 to 10) C(21, j) = 2^20</p><p><strong>∴ Answer: 2<sup>20</sup></strong></p>
Correct Answer: 2^20