Every number is the two above it, added.

Start with a 1 at the top and build down. That one rule makes Pascal's triangle — and its rows turn out to be exactly the numbers you need to multiply out $(a + b)^n$.

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Each entry is the sum of the two above it — the 6 in row 4 is $3 + 3$. The edges are always 1. Count rows from 0 at the top.

Row $n$ gives the coefficients of $(a + b)^n$. Powers of $a$ fall from $n$ to 0 while powers of $b$ rise from 0 to $n$ — every term's powers add up to $n$.

No need to build the whole triangle: $\binom{n}{r}$ is the entry $r$ places along row $n$, counting from 0. So the $x^3$ coefficient in $(1 + x)^7$ is $\binom{7}{3} = 35$.

With $x = 0.02$ each term is many times smaller than the last, so three terms already give four-figure accuracy — the true value is $1.171659\ldots$ — and a fourth closes almost all of the gap.

One rule, a whole triangle.

Coefficients inside the bracket, the binomial theorem for $(1 + x)^n$, and twelve graded exam questions with full worked solutions are waiting.

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