18 Choose 13

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

What is 18 choose 13?

Answer

18 choose 13 is
8,568

Order doesn't matter, no repeats: 8,568 ways to choose 13 of 18 items.

Choosing 13 of 18 is the same as leaving 5 out: C(18, 13) = C(18, 5).

C(18, 13) = 18! / (13! · 5!)

= C(18, 5)

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

= 8,568

If order matters (permutations): P(18, 13) = 18 × 17 × 16 × … × 6 = 53,353,114,214,400 — each combination in 13! = 6,227,020,800 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 13 for other n

n choose 13Answer
13 choose 131
14 choose 1314
15 choose 13105
16 choose 13560
17 choose 132,380
18 choose 138,568
19 choose 1327,132
20 choose 1377,520

FAQ

What is 18 choose 13?

18 choose 13 is 8,568

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

Every way to choose 13 of 18 items leaves exactly 5 behind, so choosing 13 and choosing which 5 to leave out are the same count: 18 choose 13 = 18 choose 5 = 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, 13) = 53,353,114,214,400. Each of the 8,568 combinations can be put in order in 13! = 6,227,020,800 ways, so permutations = combinations × k!: P(n, k) = n! / (n − k)!.

See Also