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Intuitively I understand something like "10 choose 3" as someone holding out an opaque bag containing 10 different objects and asking me to put my hand in and then instructing me, "from the 10 choose 3". So it's asking, how many different ways are there of me doing that. Fair enough.

But in part of the MIT opencourseware the author uses the same "10 choose 3" to calculate the number of different 10-flip coin sequences there are with exactly 3 heads.

I can't see how to relate the coin flipping to my "choose objects from a bag" intuitive example. In the coin experiment, in what way is a coin-flip equal to one of the ten objects in the bag, and in what way is each of the three heads like one of the three objects I pull out? Can anyone help? thx.

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The interpretation is that in $n$ independent trials of a coin flip, we know both the outcome and the order of the flips. So if we number the flips in the order they were made, then we have a natural assignment of "heads" or "tails" to the sequence $\{1, 2, 3, \ldots, n\}$. Then, when counting which tosses were heads (and by exclusion, which were tails), we are simply choosing which of those numbers were heads. For example, if we saw the sequence $$\{H, T, H, H, T, T, T, H, T, T\}$$ in $n = 10$ tosses, we observed $4$ heads, but the location of those four heads in the sequence is $\{1, 3, 4, 8\}$. Thus this outcome is tantamount to selecting the set $\{1, 3, 4, 8\}$ from $\{1, 2, 3, \ldots, 10\}$. And if we are interested in the number of ways to get $k$ heads out of $n$ flips, we can now see the immediate connection to binomial coefficients: it is equivalent to the number of distinct $k$-subsets that can be selected from a set of $n$ distinct numbers (or objects).

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    $\begingroup$ Ah, cool. So the "objects in the bag" could be disks, numbered 1 through 10, with each one representing "position in the sequence where the coin landed as head". Then me "choosing 3" is equivalent to noting which of the ten tosses did indeed land as head. Excellent, thanks! $\endgroup$
    – tkp
    Commented Jan 19, 2015 at 23:24

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