Q: Count the number of ways to get 3 heads in 10 flips of a coin.

A: 10!/(3!7!)

I wonder why it is a combination problem.

The post Probability of 3 Heads in 10 Coin Flips (2nd answer): "We can choose $3$ objects from $10$ in $\binom{10}{3}$ ways"

I don't really see how.

When I look at another example online: list all combination of 3 elements out of the set of {a,b,c,d}.

I can visualize it by thinking of number of possible ways for the first element is 4, number of possible ways for the 2nd element is 3, and so on. And so, it is 4 * 3 * 2.

It would be great if someone can explain what 3 objects are picked from 10 things in the coin example. What are the "objects" and what are "things".


  • 1
    $\begingroup$ You choose the three slots into which the Heads fall (the other slots are assigned to Tails). $\endgroup$ – lulu Aug 16 '17 at 17:22

Think of the "objects" as little coin holders. Each time you flip a coin, you put that coin into one of your ten little coin holders, with either a heads or tails showing. If you want to get exactly 3 heads, then you are picking three of your little coin holders to hold coins with heads showing (which forces the remaining 7 coin holders to hold coins with tails showing).


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