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15 Choose 13

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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)

= 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
k	15 choose k
0	1
1	15
2	105
3	455
4	1,365
5	3,003
6	5,005
7	6,435
8	6,435
9	5,005
10	3,003
11	1,365
12	455
13	105
14	15
15	1

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

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https://calculat.io/en/number/combinations/15--13

<a href="https://calculat.io/en/number/combinations/15--13">15 Choose 13: Combinations C(15, 13) - Calculatio</a>

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15 Choose 13: Combinations C(15, 13). 15 choose 13 = 105: the number of ways to choose 13 of 15 items when order doesn't matter and nothing repeats. C(15, 13) = (15 × 14) / (2 × 1).

https://calculat.io/en/number/combinations/15--13/generated.jpg

n choose 13 for other n

n choose 13	Answer
13 choose 13	1
14 choose 13	14
15 choose 13	105
16 choose 13	560
17 choose 13	2,380
18 choose 13	8,568
19 choose 13	27,132
20 choose 13	77,520

Recalculate with different values ↑

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)!.