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solutions7.1.txt
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Michael Klauser Exercise 7.1 We suppose 7 blue and pink balls, each of them uniquely so that we can distinguish them. We only need to calculate how often we get k successes in n trials We can now draw n! samples of balls. This sample can contain the same balls but different ordered. Because, we have n choices for the first n-1 for the second and so on. General n! but now we only want to distinguish between blue and pink balls for k blue balls, we again have k! possibilities to bring them in order similarly, for the remaining n-k balls. So we have (n-k)! possibilities. Now we only count the number of blue and pink balls. (we don't care about the order) Now we can divide the overall-number of possibilities n! by the number of possibilities for the blue balls k! and by the number of possibilities for pink balls (n-k)!. We remember that all events are statistically independent, this yields: binomial coefficient = n! / (k! (n-k)!)
