Pascal's Triangle and the Binomial Expansion: Quick Revision

A brief note on Pascal's Triangle and the Binomial Expansion. Read it before a test. Try each question at the end before you open its answer.

The expansion

• The binomial theorem: $(a + b)^n = \displaystyle\sum_{r=0}^{n} {}^nC_r\,a^{n-r}b^r$, with $n + 1$ terms.
• Pascal's triangle: row $n$ holds the coefficients. Each number is the sum of the two above it.
• Why ${}^nC_r$: it counts the ways to choose which $r$ of the $n$ brackets give a $b$.
• A difference: $(a - b)^n$ has alternating signs.
• The sum of a row: ${}^nC_0 + {}^nC_1 + \dots + {}^nC_n = 2^n$.
• The sum of the coefficients of any expansion: put $x = 1$ in it.

The general term

• The term: $T_{r+1} = {}^nC_r\,a^{n-r}b^r$, the $(r + 1)$th term.
• From the end: the $p$th term from the end is the $(n - p + 2)$th term from the start.
• A given power of $x$: write the power in terms of $r$ and match it. Solve for $r$. Check that $r$ is a whole number from $0$ to $n$.

Middle terms and the term free of $x$

• Even $n$: one middle term, $T_{n/2+1}$.
• Odd $n$: two middle terms, $T_{(n+1)/2}$ and $T_{(n+3)/2}$.
• Free of $x$: set the power of $x$ to $0$.
• A small power: for a small $x$, $(1 + x)^n \approx 1 + nx + \dfrac{n(n - 1)}{2}x^2$.

A worked example

Find the coefficient of $x^{4}$ in the expansion of $\left(x^2 + \dfrac{3}{x}\right)^{7}$.

1. The general term. $T_{r+1} = {}^7C_r\,x^{2(7-r)}\,3^r\,x^{-r} = {}^7C_r\,3^r\,x^{14-3r}$.
2. Match the power. $14 - 3r = 4$ gives $r = \dfrac{10}{3}$, which is not a whole number.
3. The answer. There is no term with $x^4$, so its coefficient is $0$. For $x^5$ instead, $r = 3$ and the coefficient is ${}^7C_3 \times 27 = 945$.

Practice with new numbers

A program to try

The program checks the formulas of this note on one expansion, $(2x + 3)^5$: the terms, their sum, the number of terms, the row of Pascal's triangle and the middle terms.

Mistakes to avoid

• Counting $n$ terms. There are $n + 1$.
• $T_r$ for the general term. The formula is the $(r + 1)$th term.
• Dropping the numbers inside the brackets. In $(2x + 3)^n$ the coefficient is ${}^nC_r\,2^{n-r}3^r$.
• Skipping the check on $r$. A fraction means there is no such term.
• One middle term when $n$ is odd. There are two.

Check yourself

1. How many terms does $(a - b)^{9}$ have?

2. Find the $4$th term of $(x + 2)^{6}$.

3. Which term of $(x + y)^{10}$ contains $y^7$?

4. Name the middle term or terms of $(a + b)^{9}$.

5. Is there a term free of $x$ in $\left(x^2 + \dfrac{1}{x}\right)^{9}$?