Combinatorics & Pascal's Triangle in 2026
The binomial coefficient $\binom{n}{k}$, pronounced "$n$ choose $k$," is the foundational building block of combinatorics, probability theory, and polynomial algebra. It indexes the coefficient of $x^k y^{n-k}$ in the algebraic expansion of $(x + y)^n$.
The fundamental symmetry theorem $\binom{n}{k} = \binom{n}{n - k}$ means choosing 3 items out of 10 leaves 7 items behind, yielding the exact same combinatorial count.
Pascal's Triangle Row Benchmark Table
| Row ($n$) | Binomial Coefficients (Row Entries) | Sum of Row ($2^n$) |
|---|---|---|
| Row 0 | 1 | 1 ($2^0$) |
| Row 1 | 1, 1 | 2 ($2^1$) |
| Row 2 | 1, 2, 1 | 4 ($2^2$) |
| Row 3 | 1, 3, 3, 1 | 8 ($2^3$) |
| Row 4 | 1, 4, 6, 4, 1 | 16 ($2^4$) |
| Row 5 | 1, 5, 10, 10, 5, 1 | 32 ($2^5$) |
Mathematical Formulations
\binom{n}{k} = \frac{n!}{k!(n - k)!} = \frac{n(n-1)(n-2)\cdots(n - k + 1)}{k!}
\text{Pascal's Identity: } \binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}