15 Choose 10

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

What is 15 choose 10?

Answer

15 choose 10 is
3,003

Order doesn't matter, no repeats: 3,003 ways to choose 10 of 15 items.

Choosing 10 of 15 is the same as leaving 5 out: C(15, 10) = C(15, 5).

C(15, 10) = 15! / (10! · 5!)

= C(15, 5)

= (15 × 14 × 13 × 12 × 11) / (5 × 4 × 3 × 2 × 1)
= 360,360 / 120

= 3,003

If order matters (permutations): P(15, 10) = 15 × 14 × 13 × … × 6 = 10,897,286,400 — each combination in 10! = 3,628,800 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 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 15 choose 10?

15 choose 10 is 3,003

Why is 15 choose 10 the same as 15 choose 5?

Every way to choose 10 of 15 items leaves exactly 5 behind, so choosing 10 and choosing which 5 to leave out are the same count: 15 choose 10 = 15 choose 5 = 3,003. In the formula n! / (k! (n − k)!) the two factorials below just swap places.

How many ways are there if order matters?

P(15, 10) = 10,897,286,400. Each of the 3,003 combinations can be put in order in 10! = 3,628,800 ways, so permutations = combinations × k!: P(n, k) = n! / (n − k)!.

See Also