Pascal's Triangle
A triangle of numbers built from nothing but addition — and hidden inside it, every coefficient you'll ever need to expand $(a+b)^n$.
Building the triangle
Pascal's triangle is a triangular array of numbers, built by one simple rule: each entry is the sum of the two entries directly above it. Every row starts and ends with $1$, since there's only ever one number above those outer positions to "carry down".
Rows are numbered starting from $\mathbf{0}$: the single $1$ at the top is row 0, then $1,1$ is row 1, then $1,2,1$ is row 2, and so on. Within a row, entries are also numbered from $0$, so row $4$ reads $1,4,6,4,1$ at positions $0,1,2,3,4$.
Think of each row as the coefficients of $(a+b)^n$ (you'll prove this properly in the next lessons). Going from row $n-1$ to row $n$ means multiplying by one more $(a+b)$. Every existing term gets hit twice — once by the new $a$, once by the new $b$ — so each new coefficient is built from two old ones added together. The addition rule isn't a curiosity; it's multiplication by $(a+b)$, written as a picture.
Two patterns worth knowing
Symmetry: every row reads the same forwards and backwards — the entry at position $r$ equals the entry at position $n-r$. Row sum: the entries of row $n$ always add up to $2^n$ — row $2$ sums to $1+2+1=4=2^2$, row $4$ sums to $1+4+6+4+1=16=2^4$, and so on.
Pascal's triangle is quick to build but slow to extend by hand — writing out row $20$ one addition at a time is painful. The next lesson introduces factorial notation and the $\binom{n}{r}$ formula, which computes any single entry of any row directly, without building the rows above it. Pascal's triangle is the picture; factorials are the shortcut.
Worked example 1
The number of entries in row $n$ is always $n+1$ (because counting starts at $0$). Row $8$ has $9$ entries — a quick way to check you haven't dropped one while adding.
Worked example 2
This pattern isn't a coincidence to memorise blindly: setting $a=b=1$ in $(a+b)^n$ gives $(1+1)^n = 2^n$ on one side, and "the sum of all the coefficients" on the other — because every power of $1$ is just $1$. Row sums and powers of $2$ are the same fact seen two ways.
Worked example 3
The number of terms in the expansion of $(x+y)^n$ is $n+1$, not $n$ — easy to forget when $n$ is odd, like here, where $5$ gives $6$ terms. Count row entries, not the power itself.
Worked example 4
Position $2$ of each row (the third number along) follows its own pattern as you move down the triangle. Building that single column, row by row, tracks down the answer without needing a formula.
The position-2 column of Pascal's triangle — $1, 3, 6, 10, 15, 21, 28, 36, 45, \ldots$ — is exactly the sequence of triangular numbers, because each one is built by adding one more than the previous increase. Pascal's triangle quietly contains other well-known number sequences inside it; spotting them is often faster than building full rows.
Miscounting which row or position you're on is the single biggest source of error in this topic — always double check by re-deriving a couple of rows from row $0$ rather than trusting a remembered row from memory.
Check yourself
Row 9 of Pascal's triangle is $1, 9, 36, 84, 126, 126, 84, 36, 9, 1$. Write down row 10, and state the sum of its entries.
(Hint: add adjacent pairs from row 9, then use $2^n$ for the sum.)