I have a book, and I dont get it. We are talking about the following situation.Lets say you have 5 experiments and the sample space for each is $S = \{0,1\}$. So the total amount of possible outcomes is $2^5 = 32$. I get that. Next up, find all outcomes with exactly 3 ones in it. The outcomes are then given: ten in total. And then the following formula is given for finding out that there are exactly ten outcomes: $$ {5 \choose 3} $$ Then everything gets generalized and, if the sample space $S$, there are $n$ subexperiments and you want to find out how many outcomes with $k$ ones there are, you do the following (given that $ k \leq n $): $$ {n \choose k} $$ But I dont get it. I can see that it is true by doing the counting and math on paper, but I cannot prove it. Why is this true? How do I prove it?
2026-03-29 15:53:30.1774799610
Sampling with replacement: finding all answers with one specific characteristic
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The formula $5 \choose 3$ counts the number of ways to choose a subset of $3$ objects from a set of $5$. If you put the letters $A,B,C,D,E$ in a bag and draw three at random, you will get one of ${5 \choose 3} = 10$ outcomes ($ABC, ABD, \ldots, CDE$). Each of those outcomes can be associated with an experiment where the letters you picked correspond to $1$s: