Binomial Coefficients: Expansion, Counting, and Identities

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Binomial theorem

For a nonnegative integer $n$, $(x+y)^n=\sum_{k=0}^{n}\binom{n}{k}x^{n-k}y^k$.

Factorial formula for a binomial coefficient

For $0\leq k\leq n$, $\binom{n}{k}=\frac{n!}{k!(n-k)!}$, with $0!=1$.

Why divide by $k!$ when counting unordered selections?

Each group of $k$ distinct objects is counted once for each of its $k!$ orderings.

Generalized binomial coefficient

For nonnegative integer $k$, $\binom{\alpha}{k}=\frac{\alpha(\alpha-1)\cdots(\alpha-k+1)}{k!}$.

Multinomial coefficient

For $k_1+\cdots+k_r=n$, $\binom{n}{k_1,\ldots,k_r}=\frac{n!}{k_1!\cdots k_r!}$; it is the coefficient of $x_1^{k_1}\cdots x_r^{k_r}$ in $(x_1+\cdots+x_r)^n$.

Quiz questions

1. For a set of $n$ distinct objects, what does $\binom{n}{k}$ count?Answer: Unordered selections of exactly $k$ objects
2. Which formula gives $\binom{n}{k}$ when $0\leq k\leq n$?Answer: $\frac{n!}{k!(n-k)!}$
3. For $1\leq k<n$, which equation is Pascal's rule?Answer: $\binom{n}{k}=\binom{n-1}{k-1}+\binom{n-1}{k}$
4. What is the sum of all coefficients in row $n$ of Pascal's triangle?Answer: $2^n$
5. For which values of $x$ does the stated Newton binomial series converge for a general exponent $\alpha$?Answer: $|x|<1$