18 Choose 16

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

What is 18 choose 16?

Answer

18 choose 16 is
153

Order doesn't matter, no repeats: 153 ways to choose 16 of 18 items.

Choosing 16 of 18 is the same as leaving 2 out: C(18, 16) = C(18, 2).

C(18, 16) = 18! / (16! · 2!)

= C(18, 2)

= (18 × 17) / (2 × 1)
= 306 / 2

= 153

If order matters (permutations): P(18, 16) = 18 × 17 × 16 × … × 3 = 3,201,186,852,864,000 — each combination in 16! = 20,922,789,888,000 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 16 for other n

n choose 16Answer
16 choose 161
17 choose 1617
18 choose 16153
19 choose 16969
20 choose 164,845

FAQ

What is 18 choose 16?

18 choose 16 is 153

Why is 18 choose 16 the same as 18 choose 2?

Every way to choose 16 of 18 items leaves exactly 2 behind, so choosing 16 and choosing which 2 to leave out are the same count: 18 choose 16 = 18 choose 2 = 153. In the formula n! / (k! (n − k)!) the two factorials below just swap places.

How many ways are there if order matters?

P(18, 16) = 3,201,186,852,864,000. Each of the 153 combinations can be put in order in 16! = 20,922,789,888,000 ways, so permutations = combinations × k!: P(n, k) = n! / (n − k)!.

See Also