16 Choose 16

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

What is 16 choose 16?

Answer

16 choose 16 is
1

Order doesn't matter, no repeats: 1 way to choose 16 of 16 items.

There is exactly one way: take all.

C(16, 16) = 16! / (16! · 0!)

= 1

0! = 1 (an empty product), so 16! cancels out completely.

If order matters (permutations): P(16, 16) = 16 × 15 × 14 × … × 1 = 20,922,789,888,000 — each combination in 16! = 20,922,789,888,000 orders.

Row 16 of Pascal's triangle
k16 choose k
01
116
2120
3560
41,820
54,368
68,008
711,440
812,870
911,440
108,008
114,368
121,820
13560
14120
1516
161

Row sum: 216 = 65,536 — all subsets of 16 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 16 choose 16?

16 choose 16 is 1

Why is 16 choose 16 the same as 16 choose 0?

Every way to choose 16 of 16 items leaves exactly 0 behind, so choosing 16 and choosing which 0 to leave out are the same count: 16 choose 16 = 16 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(16, 16) = 20,922,789,888,000. The only combination — all 16 items — 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