Number of ways to choose 3 distinct slots from 10:
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- This is equivalent to the number of integer solutions to:
- Question: A science teacher uses colored balls to demonstrate probability. She has 5 red, 4 blue, and 3 green balls, all indistinguishable within colors. In how many distinct ways can she arrange all the balls in a line so that no two green balls are adjacent?
- First, total number of non-green balls:
- Now, for each such choice, we place one green ball in each selected slot, and arrange the 9 non-green balls (5 red and 4 blue) in all possible ways.
- Number of arrangements of 5 red and 4 blue balls:
- Question: A GIS analyst is labeling 10 urban zones with binary codes of length 4 (each digit 0 or 1). What is the probability that exactly three of the zones receive a code with exactly two 1s?
