18 Choose 5

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

What is 18 choose 5?

Answer

18 choose 5 is
8,568

Order doesn't matter, no repeats: 8,568 ways to choose 5 of 18 items.

C(18, 5) = 18! / (5! · 13!)

= (18 × 17 × 16 × 15 × 14) / (5 × 4 × 3 × 2 × 1)
= 1,028,160 / 120

= 8,568

If order matters (permutations): P(18, 5) = 18 × 17 × 16 × 15 × 14 = 1,028,160 — each combination in 5! = 120 orders.

Row 18 of Pascal's triangle
k18 choose k
01
118
2153
3816
43,060
58,568
618,564
731,824
843,758
948,620
1043,758
1131,824
1218,564
138,568
143,060
15816
16153
1718
181

Row sum: 218 = 262,144 — all subsets of 18 items.

See Also

Recalculate with different values ↑

n choose 5 for other n

n choose 5Answer
6 choose 56
7 choose 521
8 choose 556
9 choose 5126
10 choose 5252
11 choose 5462
12 choose 5792
13 choose 51,287
14 choose 52,002
15 choose 53,003
16 choose 54,368
17 choose 56,188
18 choose 58,568
19 choose 511,628
20 choose 515,504

FAQ

What is 18 choose 5?

18 choose 5 is 8,568

Why is 18 choose 5 the same as 18 choose 13?

Every way to choose 5 of 18 items leaves exactly 13 behind, so choosing 5 and choosing which 13 to leave out are the same count: 18 choose 5 = 18 choose 13 = 8,568. In the formula n! / (k! (n − k)!) the two factorials below just swap places.

How many ways are there if order matters?

P(18, 5) = 1,028,160. Each of the 8,568 combinations can be put in order in 5! = 120 ways, so permutations = combinations × k!: P(n, k) = n! / (n − k)!.

See Also