Find The Probability

Here's how I like to think of this problem:

Assume that each component fails or works independently from the others. For exactly 7 out of 10 components to work, we need 3 to fail. There is a 0.1 probability that a single one fails, and it follows that there is a $0.1^3$ probability that three given components will fail. It also follows that there must be a 0.9 probability that the component will work, or a $0.9^7$ probability that seven given components will work.

So what about the $\binom{10}{3}$ ? One thing you might have noticed is that we said that three components fail, but we didn't mention WHICH three. $\binom{10}{3}$ is called a binomial coefficient, and it helps us to count the distinct ways we can choose 3 items out of about group of 10 items. It is equivalent to $\frac{10!}{3! 7!}=\frac {720}{6}=120$ . Multiplying the three parts (the two powers of probabilities and this binomial coefficient), we have our solution.

Hope this helps!

yeah no shit, formula doesnt help us understand it