7 Choose 5

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

What is 7 choose 5?

Answer

7 choose 5 is
21

Order doesn't matter, no repeats: 21 ways to choose 5 of 7 items.

Choosing 5 of 7 is the same as leaving 2 out: C(7, 5) = C(7, 2).

C(7, 5) = 7! / (5! · 2!)

= C(7, 2)

= (7 × 6) / (2 × 1)
= 42 / 2

= 21

If order matters (permutations): P(7, 5) = 7 × 6 × 5 × 4 × 3 = 2,520 — each combination in 5! = 120 orders.

Row 7 of Pascal's triangle
k7 choose k
01
17
221
335
435
521
67
71

Row sum: 27 = 128 — all subsets of 7 items.

See Also

Recalculate with different values ↑

n choose 5 for other n

n choose 5Answer
2 choose 50
5 choose 51
6 choose 56
7 choose 521
8 choose 556
9 choose 5126
10 choose 5252
11 choose 5462
12 choose 5792
13 choose 51,287
14 choose 52,002
15 choose 53,003
16 choose 54,368
17 choose 56,188
18 choose 58,568

FAQ

What is 7 choose 5?

7 choose 5 is 21

Why is 7 choose 5 the same as 7 choose 2?

Every way to choose 5 of 7 items leaves exactly 2 behind, so choosing 5 and choosing which 2 to leave out are the same count: 7 choose 5 = 7 choose 2 = 21. In the formula n! / (k! (n − k)!) the two factorials below just swap places.

How many ways are there if order matters?

P(7, 5) = 2,520. Each of the 21 combinations can be put in order in 5! = 120 ways, so permutations = combinations × k!: P(n, k) = n! / (n − k)!.

See Also