You can often solve a counting problem by working out in how many ways you can choose the required object. When doing so, there are a number of questions that are nearly always worth asking yourself.
- What am I choosing? In this example, are you choosing the $1$ or the $0$s? No - you know all about these, there are three $1$s and three $0$s. What you are choosing is the places in which the $1$s and $0$s will occur. In this case, once you have chosen the places for the $1$s you know where the $0$s go (namely, in all the remaining places), so we only have to choose places for the $1$s.
- So, you have to choose three places from six possibilities. Is repetition allowed? If you are not sure, make up a specific instance and ask whether or not it makes sense. So, could we put three $1$s in places $6,6$ and $6$? Obviously not - the three $1$s must go in different places. Note how important question 1 is - if you thought you were choosing the $1$s (rather than the places) you would certainly say that repetition is allowed because there are three of them. This would be wrong.
- Is order important? Once again, if you are not sure, ask yourself about a specific instance. For example, if I choose to put $1$s in places $1,2$ and $4$, is that the same or different from putting them in places $2,4$ and $1$? I hope it is clear that it is the same (the word is $110100$ in both cases) and therefore order is not important.
So, we have to choose $3$ places from $6$, with repetition not allowed and order not important, and the number of ways to do this is $C(6,3)$.