16 Choose 3

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

What is 16 choose 3?

Answer

16 choose 3 is
560

Order doesn't matter, no repeats: 560 ways to choose 3 of 16 items.

C(16, 3) = 16! / (3! · 13!)

= (16 × 15 × 14) / (3 × 2 × 1)
= 3,360 / 6

= 560

If order matters (permutations): P(16, 3) = 16 × 15 × 14 = 3,360 — each combination in 3! = 6 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 3 for other n

n choose 3Answer
9 choose 384
10 choose 3120
11 choose 3165
12 choose 3220
13 choose 3286
14 choose 3364
15 choose 3455
16 choose 3560
17 choose 3680
18 choose 3816
19 choose 3969
20 choose 31,140
25 choose 32,300
40 choose 39,880
100 choose 3161,700

FAQ

What is 16 choose 3?

16 choose 3 is 560

Why is 16 choose 3 the same as 16 choose 13?

Every way to choose 3 of 16 items leaves exactly 13 behind, so choosing 3 and choosing which 13 to leave out are the same count: 16 choose 3 = 16 choose 13 = 560. In the formula n! / (k! (n − k)!) the two factorials below just swap places.

How many ways are there if order matters?

P(16, 3) = 3,360. Each of the 560 combinations can be put in order in 3! = 6 ways, so permutations = combinations × k!: P(n, k) = n! / (n − k)!.

See Also