Concept preview

What you will be able to do

1. 01Use Pascal's triangle and combinations to find binomial coefficients
2. 02Expand $(a+b)^n$ with the binomial theorem
3. 03Find a specific term or coefficient using the general term

Expanding $(a+b)^{n}$ produces coefficients that form Pascal's triangle: each row starts and ends with 1, and every interior number is the sum of the two above it (Pascal's rule $_{n}C_{r} = {}_{n-1}C_{r-1} + {}_{n-1}C_{r}$). Row $n$ gives the coefficients of $(a+b)^{n}$ — row 4 is $1, 4, 6, 4, 1$, matching $(a+b)^{4}$. The entries are exactly the combinations $_{n}C_{r}$, which is why counting and algebra meet here.