6 Choose 3

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

What is 6 choose 3?

Answer

6 choose 3 is
20

Order doesn't matter, no repeats: 20 ways to choose 3 of 6 items.

C(6, 3) = 6! / (3! · 3!)

= (6 × 5 × 4) / (3 × 2 × 1)
= 120 / 6

= 20

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

Row 6 of Pascal's triangle
k6 choose k
01
16
215
320
415
56
61

Row sum: 26 = 64 — all subsets of 6 items.

See Also

Recalculate with different values ↑

n choose 3 for other n

n choose 3Answer
1 choose 30
2 choose 30
3 choose 31
4 choose 34
5 choose 310
6 choose 320
7 choose 335
8 choose 356
9 choose 384
10 choose 3120
11 choose 3165
12 choose 3220
13 choose 3286
14 choose 3364
15 choose 3455

FAQ

What is 6 choose 3?

6 choose 3 is 20

Why is 6 choose 3 the largest number in row 6?

In row 6 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 = 3, so 6 choose 3 = 20 is the largest: 6 choose 2 is only 15.

How many ways are there if order matters?

P(6, 3) = 120. Each of the 20 combinations can be put in order in 3! = 6 ways, so permutations = combinations × k!: P(n, k) = n! / (n − k)!.

See Also