# Please prove that the Probability Mass Function (in Binomial Distribution) is less than 1.

I have finished elementary probability and I know the sum of all probabilites in a data set is 1.But while reading Binomial Distribution,I encountered the formula for the Probability mass distribution :

$$f(k ; n, p)=\operatorname{Pr}(K=k)= {n \choose k} p^{k}(1-p)^{n-k}$$

Well, I know that probability is a fraction less than $$1$$, and fractions multiplied with fractions will still yield a lesser fraction as the product. But what I am confused about is that the "$${n \choose k}$$" that we multiply at the start of the formula is a positive integer, and I am confused that why shouldn't the net product be more than one?

I mean, the fraction that we get after multiplying the probabilities, wouldn't that turn greater than $$1$$ if we multiply (which is repeated addition by itself) that fraction by the positive integer we get as a result of "$${n \choose k}$$" ?

Sorry about my roundabout way of talking. You are only requested to prove that the whole thing is a value less than $$1$$. I tried and just couldn't see why it shouldn't be greater than $$1$$. I am in learner's stage (I could have learnt the forumla by rote, but my video instructor says if I proceed like that without understanding things and asking questions, then I will be like a monkey on a type-writer)

$$1 = 1^n = (p+1-p)^n = \sum_{k=0}^n {n \choose k} p^k(1-p)^{n-k}$$