AHL 1.10—Perms and combs, binomial with negative and fractional indices Notes

The Factorial

A factorial is an operation on a number that multiplies it by all the positive integers below it. More formally,

$$ n! = n \times (n-1) \times (n-2) \times ... \times 2 \times 1$$

$n!$ is also equal to the number of ways you can arrange $n$ objects in order. To illustrate this, let's consider five objects A, B, C, D, and E.

For the first object, we can choose between any of the five objects. This means that there are $5$ possibilities.

For the second object, we've already chosen one object to put in front, so we can choose between four objects for the second position. This means that there are now $5\times4$ possibilities.

We can continue this for the rest of the positions, showing that there are $5\times4\times3\times2\times1$ possibilities to arrange all $5$ objects. This is the definition of $5!$, and you can do a similar process for any $n$ objects to find the number of possibilities as $n!$.

Counting Principles: Permutations and Combinations

Permutations

Permutations are arrangements of objects where order matters. $nPr$ represents the number of possibilities if you take $r$ objects from a set of $n$, where order matters.

This is very similar to a factorial, however instead of going all the way down to $1$, we are multiplying $n\times(n-1)\times(n-2)\times...\times(n-r)$. So, as $(n-r)!$ multiplies the rest of the numbers down to $1$:

$nPr = n\times(n-1)\times(n-2)\times...\times(n-r)=\frac{n!}{(n-r)!}$

Combinations

Combinations are selections of objects where order doesn't matter.

For example, let's say we have four tennis players: Abigail, Brennan, Corey, and Dennis. You need to select two for a match, so the possibilities are:

• Abigail and Brennan
• Abigail and Corey
• Abigail and Dennis
• Brennan and Corey
• Brennan and Dennis
• Corey and Dennis

There are six possibilities. We notate this as $4C2 = 6$.

The number of combinations of $n$ objects taken $r$ at a time is:

$nCr = \binom{n}{r} = \frac{n!}{r!(n-r)!}$

You can think of this as $nPr$, divided by the number of permutations of $r$.

Binomial Theorem with Fractional and Negative Indices

The binomial theorem for positive integer exponents is:

$$(a+b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$$

This can be extended to fractional and negative indices.

Fractional Indices

For fractional indices, we use the general form:

$$(a+b)^n = a^n(1+\frac{b}{a})^n = a^n \sum_{k=0}^{\infty} \binom{n}{k} (\frac{b}{a})^k$$

where $n$ is a fraction and $|\frac{b}{a}| < 1$.

This is notated in the formula booklet as:

$$(a+b)^n = a^n(1+n(\frac{b}a+\frac{n(n-1)}{2!}(\frac{b}a)^2+...$$

which might make it easier to apply.

Negative Indices

For negative indices, the expansion is similar:

$$(a+b)^{-n} = a^{-n}(1+\frac{b}{a})^{-n} = a^{-n} \sum_{k=0}^{\infty} \binom{-n}{k} (\frac{b}{a})^k$$

Where $n$ is positive and $|\frac{b}{a}| < 1$.

Approximations Using Binomial Expansions

Binomial expansions can be used to approximate irrational numbers.