18 Choose 17

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

What is 18 choose 17?

Answer

18 choose 17 is
18

Order doesn't matter, no repeats: 18 ways to choose 17 of 18 items.

Choosing 17 of 18 is the same as leaving 1 out: C(18, 17) = C(18, 1).

C(18, 17) = 18! / (17! · 1!)

= C(18, 1)

= 18 / 1

= 18

If order matters (permutations): P(18, 17) = 18 × 17 × 16 × … × 2 = 6,402,373,705,728,000 — each combination in 17! = 355,687,428,096,000 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 17 for other n

n choose 17Answer
17 choose 171
18 choose 1718
19 choose 17171
20 choose 171,140

FAQ

What is 18 choose 17?

18 choose 17 is 18

Why is 18 choose 17 the same as 18 choose 1?

Every way to choose 17 of 18 items leaves exactly 1 behind, so choosing 17 and choosing which one to leave out are the same count: 18 choose 17 = 18 choose 1 = 18. In the formula n! / (k! (n − k)!) the two factorials below just swap places.

How many ways are there if order matters?

P(18, 17) = 6,402,373,705,728,000. Each of the 18 combinations can be put in order in 17! = 355,687,428,096,000 ways, so permutations = combinations × k!: P(n, k) = n! / (n − k)!.

See Also