13 Choose 10

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Total items (n)
Items chosen (k)

What is 13 choose 10?

Answer

13 choose 10 is
286

Order doesn't matter, no repeats: 286 ways to choose 10 of 13 items.

Choosing 10 of 13 is the same as leaving 3 out: C(13, 10) = C(13, 3).

C(13, 10) = 13! / (10! · 3!)

= C(13, 3)

= (13 × 12 × 11) / (3 × 2 × 1)
= 1,716 / 6

= 286

If order matters (permutations): P(13, 10) = 13 × 12 × 11 × … × 4 = 1,037,836,800 — each combination in 10! = 3,628,800 orders.

Row 13 of Pascal's triangle
k13 choose k
01
113
278
3286
4715
51,287
61,716
71,716
81,287
9715
10286
1178
1213
131

Row sum: 213 = 8,192 — all subsets of 13 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 13 choose 10?

13 choose 10 is 286

Why is 13 choose 10 the same as 13 choose 3?

Every way to choose 10 of 13 items leaves exactly 3 behind, so choosing 10 and choosing which 3 to leave out are the same count: 13 choose 10 = 13 choose 3 = 286. In the formula n! / (k! (n − k)!) the two factorials below just swap places.

How many ways are there if order matters?

P(13, 10) = 1,037,836,800. Each of the 286 combinations can be put in order in 10! = 3,628,800 ways, so permutations = combinations × k!: P(n, k) = n! / (n − k)!.

See Also