15 Choose 6

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

What is 15 choose 6?

Answer

15 choose 6 is
5,005

Order doesn't matter, no repeats: 5,005 ways to choose 6 of 15 items.

C(15, 6) = 15! / (6! · 9!)

= (15 × 14 × 13 × 12 × 11 × 10) / (6 × 5 × 4 × 3 × 2 × 1)
= 3,603,600 / 720

= 5,005

If order matters (permutations): P(15, 6) = 15 × 14 × 13 × 12 × 11 × 10 = 3,603,600 — each combination in 6! = 720 orders.

Row 15 of Pascal's triangle
k15 choose k
01
115
2105
3455
41,365
53,003
65,005
76,435
86,435
95,005
103,003
111,365
12455
13105
1415
151

Row sum: 215 = 32,768 — all subsets of 15 items.

See Also

Recalculate with different values ↑

n choose 6 for other n

n choose 6Answer
6 choose 61
7 choose 67
8 choose 628
9 choose 684
10 choose 6210
11 choose 6462
12 choose 6924
13 choose 61,716
14 choose 63,003
15 choose 65,005
16 choose 68,008
17 choose 612,376
18 choose 618,564
19 choose 627,132
20 choose 638,760

FAQ

What is 15 choose 6?

15 choose 6 is 5,005

Why is 15 choose 6 the same as 15 choose 9?

Every way to choose 6 of 15 items leaves exactly 9 behind, so choosing 6 and choosing which 9 to leave out are the same count: 15 choose 6 = 15 choose 9 = 5,005. In the formula n! / (k! (n − k)!) the two factorials below just swap places.

How many ways are there if order matters?

P(15, 6) = 3,603,600. Each of the 5,005 combinations can be put in order in 6! = 720 ways, so permutations = combinations × k!: P(n, k) = n! / (n − k)!.

See Also