Binomial expansion and Pascal’s triangle

What you are looking at

The coefficients in $(a+b)^n$ are the binomial coefficients $\binom{n}{r}$: $\binom{n}{r}$ counts the ways of choosing which $r$ of the $n$ brackets supply a $b$. Laid out row by row they form Pascal’s triangle, and each entry is the sum of the two above it, $\binom{n}{r} = \binom{n-1}{r-1} + \binom{n-1}{r}$. When you select a term, its entry in row $n$ turns yellow and the two entries above it are ringed in green, so the addition rule is visible on the triangle itself.

The general term is $T_{r+1} = \binom{n}{r} a^{n-r} b^{r}$. Switch to $(px+q)^n$ and the same coefficients are multiplied by powers of $p$ and $q$, so the numbers in the expansion no longer match the triangle. The working panel splits each term into its three factors, for example $\binom{5}{2} \times 2^3 \times (-3)^2 = 720$ for the $x^3$ term of $(2x-3)^5$, and states the coefficient of that power of $x$. It also shows that row $n$ adds up to $2^n$.

Try this

1. 1Tap the entry 10 at $r = 2$ in row 5. The entries 4 and 6 in row 4 are ringed, and the working shows $\binom{5}{2} = \binom{4}{1} + \binom{4}{2} = 4 + 6 = 10$.
2. 2Press Light up terms. The highlight walks along the row and then moves down to the next one, up to $n = 10$; watch the row total double each time $n$ goes up by 1.
3. 3Switch to $(px+q)^n$. At the starting values this is $(2x-3)^5 = 32x^5 - 240x^4 + 720x^3 - 1080x^2 + 810x - 243$: tap each term and notice that the signs alternate because odd powers of $-3$ are negative.
4. 4Set $p = 1$ and $q = 1$, and the coefficients of $(x+1)^n$ are exactly row $n$ of the triangle. Then press Hide formulas: row $n$ and the coefficients turn into question marks for you to predict.

Mistakes students make

• Forgetting to raise the number in front of $x$ to the power. The $x^3$ term of $(2x-3)^5$ contains $(2x)^3 = 8x^3$, not $2x^3$.
• Dropping the sign of a negative term. In $(2x-3)^5$ the second term in the bracket is $-3$, so each term carries a factor $(-3)^r$ and the signs alternate.
• Confusing the term number with $r$. The term containing $b^r$ is the $(r+1)$th term, $T_{r+1}$, and row $n$ has $n + 1$ entries because $r$ starts at 0.

Where it is taught

• IB Maths AA SL 1.9 The binomial theorem · HL 1.10 Permutations and combinations
• Edexcel A-Level Maths Pure Year 1 · Binomial expansion (positive integer powers)