15 Choose 13

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

What is 15 choose 13?

Answer

15 choose 13 is
105

Order doesn't matter, no repeats: 105 ways to choose 13 of 15 items.

Choosing 13 of 15 is the same as leaving 2 out: C(15, 13) = C(15, 2).

C(15, 13) = 15! / (13! · 2!)

= C(15, 2)

= (15 × 14) / (2 × 1)
= 210 / 2

= 105

If order matters (permutations): P(15, 13) = 15 × 14 × 13 × … × 3 = 653,837,184,000 — each combination in 13! = 6,227,020,800 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 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 15 choose 13?

15 choose 13 is 105

Why is 15 choose 13 the same as 15 choose 2?

Every way to choose 13 of 15 items leaves exactly 2 behind, so choosing 13 and choosing which 2 to leave out are the same count: 15 choose 13 = 15 choose 2 = 105. In the formula n! / (k! (n − k)!) the two factorials below just swap places.

How many ways are there if order matters?

P(15, 13) = 653,837,184,000. Each of the 105 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