15 Choose 9

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

What is 15 choose 9?

Answer

15 choose 9 is
5,005

Order doesn't matter, no repeats: 5,005 ways to choose 9 of 15 items.

Choosing 9 of 15 is the same as leaving 6 out: C(15, 9) = C(15, 6).

C(15, 9) = 15! / (9! · 6!)

= C(15, 6)

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

= 5,005

If order matters (permutations): P(15, 9) = 15 × 14 × 13 × … × 7 = 1,816,214,400 — each combination in 9! = 362,880 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 9 for other n

n choose 9Answer
9 choose 91
10 choose 910
11 choose 955
12 choose 9220
13 choose 9715
14 choose 92,002
15 choose 95,005
16 choose 911,440
17 choose 924,310
18 choose 948,620
19 choose 992,378
20 choose 9167,960

FAQ

What is 15 choose 9?

15 choose 9 is 5,005

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

Every way to choose 9 of 15 items leaves exactly 6 behind, so choosing 9 and choosing which 6 to leave out are the same count: 15 choose 9 = 15 choose 6 = 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, 9) = 1,816,214,400. Each of the 5,005 combinations can be put in order in 9! = 362,880 ways, so permutations = combinations × k!: P(n, k) = n! / (n − k)!.

See Also