<p>The number of ways of choosing 10 objects out of 31 objects of which 10 are identical and the remaining 21 are distinct, is:</p>
Step-by-Step Solution
Key Concept: Partition the selection into two cases based on how many identical objects are chosen (0 to 10), then for each case count ways to choose remaining objects from 21 distinct ones. The total is the sum of binomial coefficients.
<p><strong>Step 1:</strong> Identify the structure: 10 identical objects + 21 distinct objects. We need to choose exactly 10 objects total.</p><p><strong>Step 2:</strong> Let k = number of identical objects chosen, where k ∈ {0, 1, 2, ..., 10}.</p><p><strong>Step 3:</strong> For each value of k, we must choose (10 - k) objects from the 21 distinct objects. This can be done in C(21, 10-k) ways.</p><p><strong>Step 4:</strong> Total number of ways = Σ(k=0 to 10) C(21, 10-k)</p><p><strong>Step 5:</strong> Reindex by letting j = 10 - k, so j ranges from 10 down to 0: Σ(j=0 to 10) C(21, j)</p><p><strong>Step 6:</strong> By symmetry property: C(21,0) + C(21,1) + ... + C(21,10) = 2^20 (half of all subsets of 21 distinct objects)</p><p>∴ Answer: D (which equals 2^20 or 1,048,576)</p>
Correct Answer: D