5 Choose 3

Combination Calculator
Total items (n)
Items chosen (k)

What is 5 choose 3?

Answer

5 choose 3 is
10

Order doesn't matter, no repeats: 10 ways to choose 3 of 5 items.

Choosing 3 of 5 is the same as leaving 2 out: C(5, 3) = C(5, 2).

C(5, 3) = 5! / (3! · 2!)

= C(5, 2)

= (5 × 4) / (2 × 1)
= 20 / 2

= 10

If order matters (permutations): P(5, 3) = 5 × 4 × 3 = 60 — each combination in 3! = 6 orders.

Row 5 of Pascal's triangle
k5 choose k
01
15
210
310
45
51

Row sum: 25 = 32 — all subsets of 5 items.

See Also

Recalculate with different values ↑

n choose 3 for other n

n choose 3Answer
1 choose 30
2 choose 30
3 choose 31
4 choose 34
5 choose 310
6 choose 320
7 choose 335
8 choose 356
9 choose 384
10 choose 3120
11 choose 3165
12 choose 3220
13 choose 3286
14 choose 3364
15 choose 3455

FAQ

What is 5 choose 3?

5 choose 3 is 10

Why is 5 choose 3 the same as 5 choose 2?

Every way to choose 3 of 5 items leaves exactly 2 behind, so choosing 3 and choosing which 2 to leave out are the same count: 5 choose 3 = 5 choose 2 = 10. In the formula n! / (k! (n − k)!) the two factorials below just swap places.

How many ways are there if order matters?

P(5, 3) = 60. Each of the 10 combinations can be put in order in 3! = 6 ways, so permutations = combinations × k!: P(n, k) = n! / (n − k)!.

See Also