Virtual laboratory/Mathematics/Combinatorics/Permutations, variations, combinations

Permutations, variations, combinations

Which model should you use? Answer two questions: does order matter, and can elements repeat? Toggle the options and watch the right formula appear.

Permutations, variations, combinations
Mathematics · Combinatorics · Permutations, variations, combinations
Does order matter?
Are repetitions allowed?
Number of elements nn5
We choose kk3
When the order of the selection does not matter (we choose a subset), we count:
order
matters
repetitions
no
model
Variations
outcomes
60
🎯 Challenge: count the result
streak: 0

I’ll give you a model (order + repetitions) plus n and k — count the number of outcomes and pick the correct value.

What just happened?

What we count with is decided by two questions: does order matter? and can elements repeat? Flipping one toggle switches the model and the number of outcomes.

Combinations — order irrelevant:

We choose a subset of k elements out of n. {A,B}\{A,B\} and {B,A}\{B,A\} are the same choice, so we divide by k!k! arrangements:

Cnk=(nk)=n!k! (n−k)!C_n^k=\binom{n}{k}=\dfrac{n!}{k!\,(n-k)!}

Variations — order matters:

An ordered sequence of k elements. Without repetition each element is taken once; with repetition — any number of times:

Vnk=n!(n−k)!V‾nk=nkV_n^k=\dfrac{n!}{(n-k)!}\qquad \overline{V}_n^k=n^k

Permutations — arranging all n:

A special case of variations without repetition for k=nk=n — we order all the elements:

Pn=n!P_n=n!

Decision tree — which model to use?

Order irrelevant ⇒ combinations CnkC_n^k (no rep.) or Cn+k−1kC_{n+k-1}^{k} (with rep.). Order matters ⇒ variations: n!(n−k)!\tfrac{n!}{(n-k)!} (no rep.) or nkn^k (with rep.); and when we take all (k=nk=n) — permutations n!n!.

Why you need this for the exam:

Combinatorics problems almost always come down to asking yourself these two questions. Drawing from an urn without replacement, where order does not matter, is combinations; a PIN code (order matters, digits may repeat) is 10410^4 variations with repetition.

Remember: first ask about order, then about repetitions. Order irrelevant ⇒ combinations. These two questions settle the entire choice of model.

Illustrative model · NexTutor Virtual Laboratory

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Theory & formulas

Four models

Depending on whether order matters and whether repetitions are allowed:

  • permutations of nn elements: Pn=n!P_n=n!
  • variations without repetition (kk out of nn, order matters): n!(n−k)!\dfrac{n!}{(n-k)!}
  • variations with repetition: nkn^k
  • combinations (order does not matter): (nk)=n!k! (n−k)!\binom{n}{k}=\dfrac{n!}{k!\,(n-k)!}

How to choose

First ask about order, then about repetitions — the two answers pin down the right formula.

Notation: nn number of elements, kk number chosen, !! factorial.

Frequently asked questions

›

When do we count combinations?

When the order of the selection does NOT matter (choosing a subset), e.g. forming a committee: C(n,k).

›

How do variations differ from combinations?

In variations the order matters (we arrange the elements); in combinations it does not (we only choose them).

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