20 Choose 9

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

What is 20 choose 9?

Answer

20 choose 9 is
167,960

Order doesn't matter, no repeats: 167,960 ways to choose 9 of 20 items.

C(20, 9) = 20! / (9! · 11!)

= (20 × 19 × 18 × … × 12) / (9 × 8 × 7 × … × 1)

9 factors on top and bottom

= 60,949,324,800 / 362,880

= 167,960

If order matters (permutations): P(20, 9) = 20 × 19 × 18 × … × 12 = 60,949,324,800 — each combination in 9! = 362,880 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 9 for other n

n choose 9Answer
9 choose 91
10 choose 910
11 choose 955
12 choose 9220
13 choose 9715
14 choose 92,002
15 choose 95,005
16 choose 911,440
17 choose 924,310
18 choose 948,620
19 choose 992,378
20 choose 9167,960

FAQ

What is 20 choose 9?

20 choose 9 is 167,960

Why is 20 choose 9 the same as 20 choose 11?

Every way to choose 9 of 20 items leaves exactly 11 behind, so choosing 9 and choosing which 11 to leave out are the same count: 20 choose 9 = 20 choose 11 = 167,960. In the formula n! / (k! (n − k)!) the two factorials below just swap places.

How many ways are there if order matters?

P(20, 9) = 60,949,324,800. Each of the 167,960 combinations can be put in order in 9! = 362,880 ways, so permutations = combinations × k!: P(n, k) = n! / (n − k)!.

See Also