Lecture 3: Counting
نویسنده
چکیده
Example 3. How many ways are there to put m balls into n bins. Assume m ≤ n. 1. Balls are distinct and bins are distinct. Every ball has n choices. Hence n. Exercise 2. Why is the answer not m by looking at the opposite argument. 2. Balls are not distinct but bins are distinct. Take m identical balls and n− 1 identical sticks and permute them. Every permutation gives a different arrangement. So there are ( m+n−1 m ) ways. 3. Both are indistinguishable. Convince yourself that this is equal to number of partitions of m. We don’t have a closed form formula for this number. 4. Balls are distinct but bins are not. This is again a difficult problem. Look at Bell’s number for more information.
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