20 Choose 3

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

What is 20 choose 3?

Answer

20 choose 3 is
1,140

Order doesn't matter, no repeats: 1,140 ways to choose 3 of 20 items.

C(20, 3) = 20! / (3! · 17!)

= (20 × 19 × 18) / (3 × 2 × 1)
= 6,840 / 6

= 1,140

If order matters (permutations): P(20, 3) = 20 × 19 × 18 = 6,840 — each combination in 3! = 6 orders.

Row 20 of Pascal's triangle
k20 choose k
01
120
2190
31,140
44,845
515,504
638,760
777,520
8125,970
9167,960
10184,756
11167,960
12125,970
1377,520
1438,760
1515,504
164,845
171,140
18190
1920
201

Row sum: 220 = 1,048,576 — all subsets of 20 items.

See Also

Recalculate with different values ↑

n choose 3 for other n

n choose 3Answer
9 choose 384
10 choose 3120
11 choose 3165
12 choose 3220
13 choose 3286
14 choose 3364
15 choose 3455
16 choose 3560
17 choose 3680
18 choose 3816
19 choose 3969
20 choose 31,140
25 choose 32,300
40 choose 39,880
100 choose 3161,700

FAQ

What is 20 choose 3?

20 choose 3 is 1,140

Why is 20 choose 3 the same as 20 choose 17?

Every way to choose 3 of 20 items leaves exactly 17 behind, so choosing 3 and choosing which 17 to leave out are the same count: 20 choose 3 = 20 choose 17 = 1,140. In the formula n! / (k! (n − k)!) the two factorials below just swap places.

How many ways are there if order matters?

P(20, 3) = 6,840. Each of the 1,140 combinations can be put in order in 3! = 6 ways, so permutations = combinations × k!: P(n, k) = n! / (n − k)!.

See Also