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10 Choose 3

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What is 10 choose 3?

Answer

10 choose 3 is

120

Order doesn't matter, no repeats: 120 ways to choose 3 of 10 items.

C(10, 3) = 10! / (3! · 7!)

= 120

If order matters (permutations): P(10, 3) = 10 × 9 × 8 = 720 — each combination in 3! = 6 orders.

Row 10 of Pascal's triangle
k	10 choose k
0	1
1	10
2	45
3	120
4	210
5	252
6	210
7	120
8	45
9	10
10	1

Row sum: 2¹⁰ = 1,024 — all subsets of 10 items.

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https://calculat.io/en/number/combinations/10--3

<a href="https://calculat.io/en/number/combinations/10--3">10 Choose 3: Combinations C(10, 3) - Calculatio</a>

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10 Choose 3: Combinations C(10, 3). 10 choose 3 = 120: the number of ways to choose 3 of 10 items when order doesn't matter and nothing repeats. C(10, 3) = (10 × 9 × 8) / (3 × 2 × 1).

https://calculat.io/en/number/combinations/10--3/generated.jpg

n choose 3 for other n

n choose 3	Answer
3 choose 3	1
4 choose 3	4
5 choose 3	10
6 choose 3	20
7 choose 3	35
8 choose 3	56
9 choose 3	84
10 choose 3	120
11 choose 3	165
12 choose 3	220
13 choose 3	286
14 choose 3	364
15 choose 3	455
16 choose 3	560
17 choose 3	680

Recalculate with different values ↑

FAQ

What is 10 choose 3?

10 choose 3 is 120

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

Every way to choose 3 of 10 items leaves exactly 7 behind, so choosing 3 and choosing which 7 to leave out are the same count: 10 choose 3 = 10 choose 7 = 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, 3) = 720. Each of the 120 combinations can be put in order in 3! = 6 ways, so permutations = combinations × k!: P(n, k) = n! / (n − k)!.