Combinations & Permutations

All versionsCOMBINPERMUTFACT

How many ways can you choose or arrange things? COMBIN counts selections where order doesn’t matter (lottery picks, teams); PERMUT counts arrangements where order does (rankings, PINs); FACT is the factorial behind both.

Quick formula: to count ways to choose 3 items from 10 (order ignored):

=COMBIN(10, 3)

COMBIN(10,3) = 120. Use PERMUT(10,3) = 720 when the order of the 3 matters.

Functions used (tap for the full reference guide):

COMBIN PERMUT FACT

The example

Choosing/arranging 3 from a pool of 10.

	A	B
1	Question	Count
2	Choose 3 of 10 (order no)	120
3	Arrange 3 of 10 (order yes)	720
4	5 factorial (5!)	120

The formula

Combinations and permutations of 3 from 10:

=COMBIN(10, 3) → 120 =PERMUT(10, 3) → 720

How it works

The difference is whether order counts:

COMBIN(n, k) — the number of ways to choose k items from n when order doesn’t matter. Picking {A,B,C} is the same as {C,B,A}.
PERMUT(n, k) — the number of ways to arrange k items from n when order does matter. ABC and CBA are different.
PERMUT is always larger (it counts every ordering); here 720 vs 120 — exactly 3! (=6) times more.
FACT(n) is n! (the arrangements of all n items) — the building block: COMBIN = FACT(n)/(FACT(k)*FACT(n-k)).

Lottery odds in one formula: the chance of matching a 6-from-49 draw is =1/COMBIN(49, 6) — about 1 in 14 million. COMBIN turns “how many possible tickets” into a probability.

Try it: interactive demo

Live demo

Choose k from n; compare combinations and permutations.

n (pool)

k (choose)

COMBIN: PERMUT:

Variations

Factorial

Arrangements of all n:

=FACT(5)

Combinations with repetition

Allow repeats:

=COMBINA(n, k)

Lottery probability

Odds of one ticket:

=1 / COMBIN(49, 6)

Pitfalls & errors

#NUM! if k > n. You can’t choose more items than exist. Keep k between 0 and n.

Combinations vs permutations. If order matters (rankings, sequences), use PERMUT; if not (groups, selections), use COMBIN. Mixing them up over- or under-counts.

Big numbers overflow. FACT and PERMUT grow astronomically; beyond ~170! Excel returns errors or loses precision.

Practice workbook

📊

Download the free Combinations & Permutations practice workbook

COMBIN/PERMUT/FACT examples with results shown, the repetition and lottery-odds variants, plus 4 challenges with answers. No sign-up required.

Frequently asked questions

How do I calculate combinations in Excel?

Use =COMBIN(n, k) for the number of ways to choose k items from n when order doesn't matter, e.g. =COMBIN(10, 3) is 120.

What's the difference between COMBIN and PERMUT?

COMBIN counts selections where order is irrelevant; PERMUT counts arrangements where order matters. PERMUT(10,3)=720 is larger than COMBIN(10,3)=120.

How do I calculate lottery odds in Excel?

Use =1/COMBIN(total, picks), e.g. =1/COMBIN(49, 6) gives the probability of matching a 6-from-49 draw.

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Related formulas: Greatest common divisor & LCM · Nth root & powers · Random integer

Function references: COMBIN · PERMUT · FACT