16 Choose 13

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

What is 16 choose 13?

Answer

16 choose 13 is
560

Order doesn't matter, no repeats: 560 ways to choose 13 of 16 items.

Choosing 13 of 16 is the same as leaving 3 out: C(16, 13) = C(16, 3).

C(16, 13) = 16! / (13! · 3!)

= C(16, 3)

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

= 560

If order matters (permutations): P(16, 13) = 16 × 15 × 14 × … × 4 = 3,487,131,648,000 — each combination in 13! = 6,227,020,800 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 13 for other n

n choose 13Answer
13 choose 131
14 choose 1314
15 choose 13105
16 choose 13560
17 choose 132,380
18 choose 138,568
19 choose 1327,132
20 choose 1377,520

FAQ

What is 16 choose 13?

16 choose 13 is 560

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

Every way to choose 13 of 16 items leaves exactly 3 behind, so choosing 13 and choosing which 3 to leave out are the same count: 16 choose 13 = 16 choose 3 = 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, 13) = 3,487,131,648,000. Each of the 560 combinations can be put in order in 13! = 6,227,020,800 ways, so permutations = combinations × k!: P(n, k) = n! / (n − k)!.

See Also