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)!)