The four slip formulae

We want to draw $k$ objects from an urn of $n$. Depending on the type of “slips” and on whether repetition is allowed, four fundamental formulae are obtained.

Property — The four slip formulae

	at most once	more than once (with repetition)
numbered slips	$\dfrac{n!}{(n-k)!}$	$n^k$
blank slips	$\dbinom{n}{k} = \dfrac{n!}{k!\,(n-k)!}$	$\dbinom{n+k-1}{k}$

The formulae take traditional names:

• $\dfrac{n!}{(n-k)!}$ = simple arrangements of $n$ objects in $k$ places;
• $n^k$ = arrangements with repetition;
• $\dbinom{n}{k}$ = simple combinations (the famous “binomial coefficient”);
• $\dbinom{n+k-1}{k}$ = combinations with repetition (the “ice-cream formula”: how many cups can be made with $k$ scoops chosen among $n$ flavours, being allowed to repeat the flavours).

The special case of permutations is obtained by taking $k=n$ in the simple arrangements: $\dfrac{n!}{(n-n)!} = \dfrac{n!}{0!} = n!$.

Links

Topics: Combinatorics
Concepts: Binomial coefficient · Combinations · Arrangements · Permutations
Methods: Combinations with repetition · Simple combinations · Arrangements with repetition · Simple arrangements · Slips table
Skills: Combinatorial calculus · Using formulae