20 Choose 16

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

What is 20 choose 16?

Answer

20 choose 16 is
4,845

Order doesn't matter, no repeats: 4,845 ways to choose 16 of 20 items.

Choosing 16 of 20 is the same as leaving 4 out: C(20, 16) = C(20, 4).

C(20, 16) = 20! / (16! · 4!)

= C(20, 4)

= (20 × 19 × 18 × 17) / (4 × 3 × 2 × 1)
= 116,280 / 24

= 4,845

If order matters (permutations): P(20, 16) = 20 × 19 × 18 × … × 5 = 101,370,917,007,360,000 — each combination in 16! = 20,922,789,888,000 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 ↑

n choose 16 for other n

n choose 16Answer
16 choose 161
17 choose 1617
18 choose 16153
19 choose 16969
20 choose 164,845

FAQ

What is 20 choose 16?

20 choose 16 is 4,845

Why is 20 choose 16 the same as 20 choose 4?

Every way to choose 16 of 20 items leaves exactly 4 behind, so choosing 16 and choosing which 4 to leave out are the same count: 20 choose 16 = 20 choose 4 = 4,845. In the formula n! / (k! (n − k)!) the two factorials below just swap places.

How many ways are there if order matters?

P(20, 16) = 101,370,917,007,360,000. Each of the 4,845 combinations can be put in order in 16! = 20,922,789,888,000 ways, so permutations = combinations × k!: P(n, k) = n! / (n − k)!.

See Also