We show how Binomial coefficients help simplify expressions. We notice that these coefficients are heavily used in probability calculus. On the other hand, the Binomial theorem serves in computing powers of numbers.
The expression of Binomial Coefficients
For natural numbers
The binomial coefficients usually appear in elementary probability.
Exercise: For which integers
Proof: To prove this, we first compute
We have
The Binomial theorem
The binomial theorem has many applications in algebra, calculus, and probability. For example, it enters into the definition of the Binomial distribution in probability theory. Here we show a nice application of this Theorem.
Binomial Theorem: For
The proof of the above sum formula is based on the fact that the multiplication operation in the set of real numbers is commutative, that is
Exercise: Compute the sum
Exercise: Prove that