10 Choose 10

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

What is 10 choose 10?

Answer

10 choose 10 is
1

Order doesn't matter, no repeats: 1 way to choose 10 of 10 items.

There is exactly one way: take all.

C(10, 10) = 10! / (10! · 0!)

= 1

0! = 1 (an empty product), so 10! cancels out completely.

If order matters (permutations): P(10, 10) = 10 × 9 × 8 × … × 1 = 3,628,800 — each combination in 10! = 3,628,800 orders.

Row 10 of Pascal's triangle
k10 choose k
01
110
245
3120
4210
5252
6210
7120
845
910
101

Row sum: 210 = 1,024 — all subsets of 10 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 10 choose 10?

10 choose 10 is 1

Why is 10 choose 10 the same as 10 choose 0?

Every way to choose 10 of 10 items leaves exactly 0 behind, so choosing 10 and choosing which 0 to leave out are the same count: 10 choose 10 = 10 choose 0 = 1. In the formula n! / (k! (n − k)!) the two factorials below just swap places.

How many ways are there if order matters?

P(10, 10) = 3,628,800. The only combination — all 10 items — can be put in order in 10! = 3,628,800 ways, so permutations = combinations × k!: P(n, k) = n! / (n − k)!.

See Also