10 Choose 7

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

What is 10 choose 7?

Answer

10 choose 7 is
120

Order doesn't matter, no repeats: 120 ways to choose 7 of 10 items.

Choosing 7 of 10 is the same as leaving 3 out: C(10, 7) = C(10, 3).

C(10, 7) = 10! / (7! · 3!)

= C(10, 3)

= (10 × 9 × 8) / (3 × 2 × 1)
= 720 / 6

= 120

If order matters (permutations): P(10, 7) = 10 × 9 × 8 × … × 4 = 604,800 — each combination in 7! = 5,040 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 7 for other n

n choose 7Answer
7 choose 71
8 choose 78
9 choose 736
10 choose 7120
11 choose 7330
12 choose 7792
13 choose 71,716
14 choose 73,432
15 choose 76,435
16 choose 711,440
17 choose 719,448
18 choose 731,824
19 choose 750,388
20 choose 777,520

FAQ

What is 10 choose 7?

10 choose 7 is 120

Why is 10 choose 7 the same as 10 choose 3?

Every way to choose 7 of 10 items leaves exactly 3 behind, so choosing 7 and choosing which 3 to leave out are the same count: 10 choose 7 = 10 choose 3 = 120. In the formula n! / (k! (n − k)!) the two factorials below just swap places.

How many ways are there if order matters?

P(10, 7) = 604,800. Each of the 120 combinations can be put in order in 7! = 5,040 ways, so permutations = combinations × k!: P(n, k) = n! / (n − k)!.

See Also