16 Choose 10

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

What is 16 choose 10?

Answer

16 choose 10 is
8,008

Order doesn't matter, no repeats: 8,008 ways to choose 10 of 16 items.

Choosing 10 of 16 is the same as leaving 6 out: C(16, 10) = C(16, 6).

C(16, 10) = 16! / (10! · 6!)

= C(16, 6)

= (16 × 15 × 14 × 13 × 12 × 11) / (6 × 5 × 4 × 3 × 2 × 1)
= 5,765,760 / 720

= 8,008

If order matters (permutations): P(16, 10) = 16 × 15 × 14 × … × 7 = 29,059,430,400 — each combination in 10! = 3,628,800 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 10 for other n

n choose 10Answer
2 choose 100
10 choose 101
11 choose 1011
12 choose 1066
13 choose 10286
14 choose 101,001
15 choose 103,003
16 choose 108,008
17 choose 1019,448
18 choose 1043,758
19 choose 1092,378
20 choose 10184,756
100 choose 1017,310,309,456,440

FAQ

What is 16 choose 10?

16 choose 10 is 8,008

Why is 16 choose 10 the same as 16 choose 6?

Every way to choose 10 of 16 items leaves exactly 6 behind, so choosing 10 and choosing which 6 to leave out are the same count: 16 choose 10 = 16 choose 6 = 8,008. In the formula n! / (k! (n − k)!) the two factorials below just swap places.

How many ways are there if order matters?

P(16, 10) = 29,059,430,400. Each of the 8,008 combinations can be put in order in 10! = 3,628,800 ways, so permutations = combinations × k!: P(n, k) = n! / (n − k)!.

See Also