Out of 13 objects, 4 are indistinguishable and rest are distinct. The number of ways we can choose 4 objects out of 13 objects is __________.
Explanation:
Case 1: All 4 objects are distinct. Number of ways of choosing 4 distinct objects out of 9 = 9C4. ∴ Total number of such ways = 1 way.
Case 2: 3 objects are distinct and 1 is indistinguishable. Number of ways of choosing 3 distinct objects out of 9 = 9C3. Number of ways of choosing 1 indistinguishable object out of 4 = 1. ∴ Total number of such ways = 9C3 × 1 = 9C3.
Case 3: 2 objects are distinct and 2 are indistinguishable. Number of ways of choosing 2 distinct objects out of 9 = 9C2. Number of ways of choosing 2 indistinguishable object outs of 4 = 1. ∴ Total number of such ways = 9C2 × 1 = 9C2.
Case 4: 1 object is distinct and 3 are indistinguishable. Number of ways of choosing 1 distinct objects out of 9 = 9C1. Number of ways of choosing 3 indistinguishable objects out of 4 = 1. ∴ Total number of such ways = 9C1 × 1 = 9C1.
Case 5: All 4 are indistinguishable. Number of ways of choosing 0 distinct objects out of 9 = 9C0. Number of ways of choosing 4 indistinguishable objects out of 4 = 1. ∴ Total number of such ways = 9C1 × 1 = 9C1.
⇒ Total number of ways = 9C4 + 9C3 + 9C2 + 9C1 + 9C0 = 256
Hence, 256.
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