10 Choose 6

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

What is 10 choose 6?

Answer

10 choose 6 is
210

Order doesn't matter, no repeats: 210 ways to choose 6 of 10 items.

Choosing 6 of 10 is the same as leaving 4 out: C(10, 6) = C(10, 4).

C(10, 6) = 10! / (6! · 4!)

= C(10, 4)

= (10 × 9 × 8 × 7) / (4 × 3 × 2 × 1)
= 5,040 / 24

= 210

If order matters (permutations): P(10, 6) = 10 × 9 × 8 × 7 × 6 × 5 = 151,200 — each combination in 6! = 720 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 6 for other n

n choose 6Answer
6 choose 61
7 choose 67
8 choose 628
9 choose 684
10 choose 6210
11 choose 6462
12 choose 6924
13 choose 61,716
14 choose 63,003
15 choose 65,005
16 choose 68,008
17 choose 612,376
18 choose 618,564
19 choose 627,132
20 choose 638,760

FAQ

What is 10 choose 6?

10 choose 6 is 210

Why is 10 choose 6 the same as 10 choose 4?

Every way to choose 6 of 10 items leaves exactly 4 behind, so choosing 6 and choosing which 4 to leave out are the same count: 10 choose 6 = 10 choose 4 = 210. In the formula n! / (k! (n − k)!) the two factorials below just swap places.

How many ways are there if order matters?

P(10, 6) = 151,200. Each of the 210 combinations can be put in order in 6! = 720 ways, so permutations = combinations × k!: P(n, k) = n! / (n − k)!.

See Also