Pascal's triangle and Newton's binomial

Pascal's triangle is a triangular array where the entry in the $n$th row and $k$th position is the binomial coefficient $\binom{n}{k}$. Newton's binomial theorem states that $(a+b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$. The coefficients in the expansion are exactly the numbers in the $n$th row of Pascal's triangle.

Important properties

• Each entry is the sum of the two entries above it: $\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}$.

• The triangle is symmetric: $\binom{n}{k} = \binom{n}{n-k}$.

• The sum of the entries in the $n$th row is $2^n$.

• Binomial coefficients count the number of ways to choose $k$ objects from $n$.