8.4 Binomial Coefficients Using Pascal's Triangle

8.4 Binomial Coefficients Using Pascal's Triangle

Pascal's triangle: row $n=0$ is $1$; each row starts/ends with $1$; interior entries are sums of the two above.

Pascal's rule: $\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}$ for $0 < k < n$.

Row $n$ is $\binom{n}{0}, \binom{n}{1}, \ldots, \binom{n}{n}$.

Example 22

$(x+y)^4$ from row $1\,4\,6\,4\,1$: $x^4 + 4x^3 y + 6x^2 y^2 + 4xy^3 + y^4$.

Example 23

$(x-2)^5 = x^5 - 10x^4 + 40x^3 - 80x^2 + 80x - 32$ (coefficients $1\,5\,10\,10\,5\,1$ from Pascal).