Counting principles and the extended binomial theorem (HL)

Q: What is the Sum Rule in counting principles?

If task A can be done in \(m\) ways and task B in \(n\) ways, and the tasks are mutually exclusive, then either A or B can be done in \(m + n\) ways.

Q: What is a permutation and how is it calculated?

A permutation is an arrangement where order matters ; it is calculated using \(P(n, r) = \frac{n!}{(n-r)!}\).

Q: What is a combination and how is it calculated?

A combination is a selection where order does not matter ; it is calculated using \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\).

Q: How do you handle counting problems where specific items must be together?

Treat the items that must stay together as a single 'block' and arrange the blocks first, then multiply by the internal arrangements of the items within that block.

Q: How do you handle counting problems where specific items must be kept separate?

Arrange the unrestricted items first, then place the restricted items in the 'gaps' between them.

Q: What is the condition for convergence in the extended binomial expansion of \((1+x)^n\)?

The expansion is valid only if \(|x| < 1\).

Q: How do you expand \((a+b)^n\) using the extended binomial theorem when \(a \neq 1\)?

Factor out \(a^n\) to rewrite it as \(a^n(1 + \frac{b}{a})^n\), then expand using the standard formula with \(x = \frac{b}{a}\).

Q: How do you calculate binomial coefficients \(\binom{n}{r}\) when \(n\) is not an integer?

Use the product formula: \(\frac{n(n-1)(n-2)\dots(n-r+1)}{r!}\).

Mathematics - Analysis and Approaches · IB Diploma Programme (DP) - SL & HL · 4 min read · Updated: 7 Sept 2026

Introduction to Advanced Counting and Binomials

Welcome to one of the most practical and fascinating parts of the Higher Level (HL) syllabus! In this chapter, we explore how to count complex arrangements without actually listing them and how to expand algebraic expressions when the power is a fraction or a negative number. These tools are essential for probability, calculus, and even computer science. Don't worry if the formulas look a bit intimidating at first; we will break them down step-by-step.

Counting Principles: Permutations and Combinations

Counting in HL Mathematics isn't about "1, 2, 3..." It is about calculating the number of ways events can happen. There are two fundamental rules to remember:
1. The Product Rule: If one task can be done in \(m\) ways and a second task in \(n\) ways, then both tasks can be done in \(m \times n\) ways. (Think: "I have 3 shirts and 4 pairs of pants; I have \(3 \times 4 = 12\) outfits.")
2. The Sum Rule: If one task can be done in \(m\) ways and a second mutually exclusive task in \(n\) ways, then either the first or the second can be done in \(m + n\) ways.

Permutations (Order Matters)

A permutation is an arrangement of items where the order is important. For example, the race results "Alice 1st, Bob 2nd" is a different permutation than "Bob 1st, Alice 2nd."
The number of ways to arrange \(r\) objects from a total of \(n\) distinct objects is given by:

\(P(n, r) = \text{ }^{n}P_{r} = \frac{n!}{(n-r)!}\)

Note: \(n!\) (n factorial) means \(n \times (n-1) \times (n-2) \times \dots \times 1\).

Combinations (Order Does NOT Matter)

A combination is a selection of items where the order is irrelevant. If you are picking a team of 3 people from a group of 10, it doesn't matter if you pick Alice first or last; she is still on the team.
The number of ways to choose \(r\) objects from \(n\) is:

\(C(n, r) = \text{ }^{n}C_{r} = \binom{n}{r} = \frac{n!}{r!(n-r)!}\)

Solving Complex Counting Problems

IB exam questions often add "constraints." Here is how to handle them:

Items must be together: Tie the items together and treat them as a "single block." Arrange the blocks first, then arrange the items inside the block.
Items must be separate: Arrange the other items first, then place the restricted items in the "gaps" between them.

Quick Review: Use \(nPr\) for arrangements (keywords: arrange, line up, schedule). Use \(nCr\) for selections (keywords: choose, committee, set).

The Extended Binomial Theorem

In the SL core, you learned to expand \((a+b)^n\) where \(n\) is a positive integer using Pascal's Triangle. In HL, we extend this to cases where \(n\) is negative or a fraction (rational).

The Standard Form

The extended version is easiest to use when the expression is in the form \((1+x)^n\).

\((1+x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \dots\)

This is an infinite series. Unlike the integer version which stops, this expansion goes on forever!

Wait, is it always valid?

Because the series is infinite, we need to make sure the terms don't just keep getting bigger and bigger. For the series to "converge" (add up to a finite number), the value of \(x\) must be small.
The Condition: The expansion of \((1+x)^n\) is only valid if \(|x| < 1\).

Dealing with \((a+b)^n\)

If your expression isn't in the form \((1+x)\), you must force it to be by factoring out the \(a\):

\((a+b)^n = a^n(1 + \frac{b}{a})^n\)

In this case, the expansion is valid only if \(|\frac{b}{a}| < 1\).

Step-by-Step Example: Expanding \((1+x)^{-1}\)

1. Identify \(n\): here \(n = -1\).
2. Apply the formula: \(1 + (-1)x + \frac{(-1)(-2)}{2!}x^2 + \frac{(-1)(-2)(-3)}{3!}x^3 + \dots\)
3. Simplify: \(1 - x + x^2 - x^3 + \dots\)
4. State validity: Valid for \(|x| < 1\).

Common Mistakes to Avoid

Factorial Confusion: Remember that \(\binom{n}{r}\) for fractions isn't done with the factorial button on your calculator. You must use the formula \( \frac{n(n-1)(n-2)\dots}{r!} \). For example, if \(n = 0.5\), then the second coefficient is \(\frac{0.5(0.5-1)}{2}\).
Forgetting the \(a^n\): When expanding \((2+x)^{-2}\), students often forget to factor out the \(2^{-2}\) (which is \(1/4\)) before starting the expansion.
Signs: Be extremely careful with negative signs when \(x\) is negative, e.g., \((1-2x)^{-3}\). Treat the "x" in the formula as \((-2x)\).

Key Takeaways

1. Permutations: Order matters! \(nPr\).
2. Combinations: Order doesn't matter! \(nCr\).
3. Extended Binomial: Works for negative and fractional powers.
4. Convergence: Always check that the absolute value of the "x" term is less than 1.
5. Formatting: Always rewrite expressions into the \((1 + \text{something})^n\) format before using the AHL binomial formula.

Note: For further applications of these expansions, you may eventually see them linked to Maclaurin series in Topic 5, but for now, focus on the algebraic manipulation.

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What is the Product Rule in counting principles?

If one task can be done in \(m\) ways and a second task in \(n\) ways, then both tasks can be done in \(m \times n\) ways.

What is the Sum Rule in counting principles?

If task A can be done in \(m\) ways and task B in \(n\) ways, and the tasks are mutually exclusive, then either A or B can be done in \(m + n\) ways.

What is a permutation and how is it calculated?

A permutation is an arrangement where order matters; it is calculated using \(P(n, r) = \frac{n!}{(n-r)!}\).

What is a combination and how is it calculated?

A combination is a selection where order does not matter; it is calculated using \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\).

How do you handle counting problems where specific items must be together?

Treat the items that must stay together as a single 'block' and arrange the blocks first, then multiply by the internal arrangements of the items within that block.

How do you handle counting problems where specific items must be kept separate?

Arrange the unrestricted items first, then place the restricted items in the 'gaps' between them.

What is the general formula for the extended binomial theorem?

\((1+x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \dots\)

What is the condition for convergence in the extended binomial expansion of \((1+x)^n\)?

The expansion is valid only if \(|x| < 1\).

How do you expand \((a+b)^n\) using the extended binomial theorem when \(a \neq 1\)?

Factor out \(a^n\) to rewrite it as \(a^n(1 + \frac{b}{a})^n\), then expand using the standard formula with \(x = \frac{b}{a}\).

How do you calculate binomial coefficients \(\binom{n}{r}\) when \(n\) is not an integer?

Use the product formula: \(\frac{n(n-1)(n-2)\dots(n-r+1)}{r!}\).

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