20 Choose 6

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

What is 20 choose 6?

Answer

20 choose 6 is
38,760

Order doesn't matter, no repeats: 38,760 ways to choose 6 of 20 items.

C(20, 6) = 20! / (6! · 14!)

= (20 × 19 × 18 × 17 × 16 × 15) / (6 × 5 × 4 × 3 × 2 × 1)
= 27,907,200 / 720

= 38,760

If order matters (permutations): P(20, 6) = 20 × 19 × 18 × 17 × 16 × 15 = 27,907,200 — each combination in 6! = 720 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 6 for other n

n choose 6Answer
6 choose 61
7 choose 67
8 choose 628
9 choose 684
10 choose 6210
11 choose 6462
12 choose 6924
13 choose 61,716
14 choose 63,003
15 choose 65,005
16 choose 68,008
17 choose 612,376
18 choose 618,564
19 choose 627,132
20 choose 638,760

FAQ

What is 20 choose 6?

20 choose 6 is 38,760

Why is 20 choose 6 the same as 20 choose 14?

Every way to choose 6 of 20 items leaves exactly 14 behind, so choosing 6 and choosing which 14 to leave out are the same count: 20 choose 6 = 20 choose 14 = 38,760. In the formula n! / (k! (n − k)!) the two factorials below just swap places.

How many ways are there if order matters?

P(20, 6) = 27,907,200. Each of the 38,760 combinations can be put in order in 6! = 720 ways, so permutations = combinations × k!: P(n, k) = n! / (n − k)!.

See Also