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9 Choose 3

Combination Calculator

What is 9 choose 3?

Answer

9 choose 3 is

84

Order doesn't matter, no repeats: 84 ways to choose 3 of 9 items.

C(9, 3) = 9! / (3! · 6!)

= 84

If order matters (permutations): P(9, 3) = 9 × 8 × 7 = 504 — each combination in 3! = 6 orders.

Row 9 of Pascal's triangle
k	9 choose k
0	1
1	9
2	36
3	84
4	126
5	126
6	84
7	36
8	9
9	1

Row sum: 2⁹ = 512 — all subsets of 9 items.

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https://calculat.io/en/number/combinations/9--3

<a href="https://calculat.io/en/number/combinations/9--3">9 Choose 3: Combinations C(9, 3) - Calculatio</a>

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9 Choose 3: Combinations C(9, 3). 9 choose 3 = 84: the number of ways to choose 3 of 9 items when order doesn't matter and nothing repeats. C(9, 3) = (9 × 8 × 7) / (3 × 2 × 1).

https://calculat.io/en/number/combinations/9--3/generated.jpg

n choose 3 for other n

n choose 3	Answer
2 choose 3	0
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

Recalculate with different values ↑

FAQ

What is 9 choose 3?

9 choose 3 is 84

Why is 9 choose 3 the same as 9 choose 6?

Every way to choose 3 of 9 items leaves exactly 6 behind, so choosing 3 and choosing which 6 to leave out are the same count: 9 choose 3 = 9 choose 6 = 84. In the formula n! / (k! (n − k)!) the two factorials below just swap places.

How many ways are there if order matters?

P(9, 3) = 504. Each of the 84 combinations can be put in order in 3! = 6 ways, so permutations = combinations × k!: P(n, k) = n! / (n − k)!.