10 Choose 5

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

What is 10 choose 5?

Answer

10 choose 5 is
252

Order doesn't matter, no repeats: 252 ways to choose 5 of 10 items.

C(10, 5) = 10! / (5! · 5!)

= (10 × 9 × 8 × 7 × 6) / (5 × 4 × 3 × 2 × 1)
= 30,240 / 120

= 252

If order matters (permutations): P(10, 5) = 10 × 9 × 8 × 7 × 6 = 30,240 — each combination in 5! = 120 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 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 10 choose 5?

10 choose 5 is 252

Why is 10 choose 5 the largest number in row 10?

In row 10 of Pascal's triangle the numbers grow towards the middle and shrink after it, because n choose k = n choose (n − k). With an even n the middle is k = 5, so 10 choose 5 = 252 is the largest: 10 choose 4 is only 210.

How many ways are there if order matters?

P(10, 5) = 30,240. Each of the 252 combinations can be put in order in 5! = 120 ways, so permutations = combinations × k!: P(n, k) = n! / (n − k)!.

See Also