4.1 Binomial expansion for positive integer n; n! and nCr — Learn

Prerequisites: multiplying out two brackets; collecting like terms; the laws of indices from Section 2.1; substituting into a formula.

By the end of this section you can: work out a factorial and a binomial coefficient; expand $(a + bx)^n$ for a positive whole number $n$; find a single term or coefficient without expanding the whole bracket; and use the expansion to work out a compound-interest value.

1. Factorials

The factorial of a whole number is the product of all the whole numbers from $1$ up to that number. We write it with an exclamation mark.

The pattern continues: $6! = 6 \times 5! = 720$. Each factorial is the number times the factorial before it. By agreement, $0! = 1$, which keeps the pattern and the algebra working.

To work out $7!$ from $4!$ you do not multiply all seven numbers. Instead,

Cancelling like this is the reason factorials are useful in the next section.

2. The binomial coefficient

A binomial coefficient counts the number of ways to choose $r$ items from $n$ items, when the order does not matter. We write it as $\binom{n}{r}$, and also as $^nC_r$.

Example 1. Work out $\binom{7}{3}$.

The $4!$ cancels because it appears on the top and the bottom. This is why you rarely need a full factorial.

Two properties follow from the definition and save work:

The second property says that choosing $3$ items to keep is the same as choosing the other $4$ to leave behind. For example, $\binom{7}{3} = \binom{7}{4} = 35$.

3. Pascal's triangle

Write the binomial coefficients in rows, one row for each $n$, and a pattern appears.

Each entry is the sum of the two entries above it. The row for $n$ reads $\binom{n}{0}, \binom{n}{1}, \dots, \binom{n}{n}$. The triangle is symmetric, which is the property $\binom{n}{r} = \binom{n}{n-r}$ again.

4. The binomial theorem

The expansion of $(a + b)^n$ for a positive whole number $n$ is

The powers of $a$ fall from $n$ to $0$ and the powers of $b$ rise from $0$ to $n$. The coefficients are the entries of row $n$ of Pascal's triangle.

Example 2. Expand $(1 + 2x)^5$.

Here $a = 1$ and $b = 2x$, and $n = 5$. The coefficients are $1, 5, 10, 10, 5, 1$.

Work out each power of $2x$ separately:

The power of $x$ matches the power of $b$ in each term. This is the quickest check on a long expansion.

Example 3. Expand $(2 - 3x)^4$.

Now $a = 2$ and $b = -3x$. Take care with the signs, because every odd power of $-3x$ is negative.

Work out each term:

The signs alternate. The constant term is $2^4 = 16$, which checks the first term.

5. One term at a time

Often you need one term, and expanding everything is wasted work. The term in $x^r$ in the expansion of $(a + bx)^n$ is

Example 4. Find the coefficient of $x^3$ in the expansion of $(2 + 5x)^6$.

Use $n = 6$, $r = 3$, $a = 2$ and $b = 5$:

The term is $20{,}000x^3$. Working out one coefficient takes one line, so there is no need to expand the other six terms.

6. A link to probability

The coefficient $\binom{n}{r}$ counts choices. Toss a coin five times. The number of orders in which exactly two heads can appear is $\binom{5}{2} = 10$. So the probability of exactly two heads is $\frac{10}{32}$, because there are $2^5 = 32$ equally likely orders in total. This is the binomial distribution, and Chapter 13 returns to it.

7. Try it: the binomial row explorer

Predict first. The figure shows a row of Pascal's triangle and one term of $(1 + x)^n$. Before you touch the controls, predict what happens to the coefficient when $r$ moves from $2$ to $n - 2$, for a fixed $n$. Write your prediction down.

Test. Move the slider for $n$ and watch the row change. Then keep $n$ fixed and move the slider for the term number $r$. The readout gives the coefficient $\binom{n}{r}$ and the term it produces.

Explain. For one setting, compare the coefficient at $r$ with the coefficient at $n - r$. Write two sentences saying why those two coefficients are equal. Use the idea of choosing which items to keep and which to leave behind.

Watch out. The commonest error is to forget the power of $a$. In Example 3 the term in $x^2$ of $(2 - 3x)^4$ is $\binom{4}{2}2^2(-3x)^2 = 6 \times 4 \times 9x^2 = 216x^2$. Writing only $\binom{4}{2}(-3x)^2$ gives $54x^2$, which drops the factor $2^2$. Re-check: put $x = 0$ into your expansion. Every term with an $x$ disappears, and what is left must be the constant term $a^n$. For $(2 - 3x)^4$ that is $16$, as it should be.

Progress check

Return to the chapter problem. The former resident gives £2,000, and the account pays 6% each year.

1. Write the value after one year as $2000 \times 1.06$.
2. Expand $(1 + 0.06)^3$ with the binomial theorem, using $n = 3$.
3. Work out the value of the gift after 3 years, to the nearest penny.

You should now be able to

• work out a factorial, and cancel factorials before multiplying;
• work out a binomial coefficient with $\binom{n}{r} = \dfrac{n!}{r!\,(n-r)!}$;
• use the symmetry $\binom{n}{r} = \binom{n}{n-r}$;
• expand $(a + bx)^n$ for a positive whole number $n$, with the correct signs;
• find a single term or coefficient without expanding the whole bracket;
• use a binomial expansion to work out a compound-interest value.