20 Choose 20

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

What is 20 choose 20?

Answer

20 choose 20 is
1

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

There is exactly one way: take all.

C(20, 20) = 20! / (20! · 0!)

= 1

0! = 1 (an empty product), so 20! cancels out completely.

If order matters (permutations): P(20, 20) = 20 × 19 × 18 × … × 1 = 2,432,902,008,176,640,000 — each combination in 20! = 2,432,902,008,176,640,000 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 ↑

FAQ

What is 20 choose 20?

20 choose 20 is 1

Why is 20 choose 20 the same as 20 choose 0?

Every way to choose 20 of 20 items leaves exactly 0 behind, so choosing 20 and choosing which 0 to leave out are the same count: 20 choose 20 = 20 choose 0 = 1. In the formula n! / (k! (n − k)!) the two factorials below just swap places.

How many ways are there if order matters?

P(20, 20) = 2,432,902,008,176,640,000. The only combination — all 20 items — can be put in order in 20! = 2,432,902,008,176,640,000 ways, so permutations = combinations × k!: P(n, k) = n! / (n − k)!.

See Also