20 Choose 5

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

What is 20 choose 5?

Answer

20 choose 5 is
15,504

Order doesn't matter, no repeats: 15,504 ways to choose 5 of 20 items.

C(20, 5) = 20! / (5! · 15!)

= (20 × 19 × 18 × 17 × 16) / (5 × 4 × 3 × 2 × 1)
= 1,860,480 / 120

= 15,504

If order matters (permutations): P(20, 5) = 20 × 19 × 18 × 17 × 16 = 1,860,480 — each combination in 5! = 120 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 5 for other n

n choose 5Answer
6 choose 56
7 choose 521
8 choose 556
9 choose 5126
10 choose 5252
11 choose 5462
12 choose 5792
13 choose 51,287
14 choose 52,002
15 choose 53,003
16 choose 54,368
17 choose 56,188
18 choose 58,568
19 choose 511,628
20 choose 515,504

FAQ

What is 20 choose 5?

20 choose 5 is 15,504

Why is 20 choose 5 the same as 20 choose 15?

Every way to choose 5 of 20 items leaves exactly 15 behind, so choosing 5 and choosing which 15 to leave out are the same count: 20 choose 5 = 20 choose 15 = 15,504. In the formula n! / (k! (n − k)!) the two factorials below just swap places.

How many ways are there if order matters?

P(20, 5) = 1,860,480. Each of the 15,504 combinations can be put in order in 5! = 120 ways, so permutations = combinations × k!: P(n, k) = n! / (n − k)!.

See Also