IB Maths AA 1.7 Notes

Factorials and binomial coefficients

The binomial theorem gives a systematic way to expand expressions of the form $(a+b)^n$, where $n\in\mathbb{N}$. It allows us to find every term in the expansion, identify coefficients, and determine specific terms without expanding everything manually. Before using the theorem, it is important to understand factorials and binomial coefficients.

Factorial notation is written as $n!$ and means the product of all positive integers from $n$ down to $1$.

For example, $5!=5\times4\times3\times2\times1=120$. Also, by definition, $0!=1$.

The binomial coefficient is written as ${}^nC_r$ or $\binom{n}{r}$ and is given by

${}^nC_r=\frac{n!}{r!(n-r)!}$

It counts the number of ways of choosing $r$ objects from $n$ objects when order does not matter.

Solve ${}^5C_2$

${}^5C_2=\frac{5!}{2!(5-2)!}=\frac{5!}{2!3!}=\frac{5\times4\times3!}{2\times1\times3!}=\frac{20}{2}=10$

So ${}^5C_2=10$.