17 Choose 3

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

What is 17 choose 3?

Answer

17 choose 3 is
680

Order doesn't matter, no repeats: 680 ways to choose 3 of 17 items.

C(17, 3) = 17! / (3! · 14!)

= (17 × 16 × 15) / (3 × 2 × 1)
= 4,080 / 6

= 680

If order matters (permutations): P(17, 3) = 17 × 16 × 15 = 4,080 — each combination in 3! = 6 orders.

Row 17 of Pascal's triangle
k17 choose k
01
117
2136
3680
42,380
56,188
612,376
719,448
824,310
924,310
1019,448
1112,376
126,188
132,380
14680
15136
1617
171

Row sum: 217 = 131,072 — all subsets of 17 items.

See Also

Recalculate with different values ↑

n choose 3 for other n

n choose 3Answer
9 choose 384
10 choose 3120
11 choose 3165
12 choose 3220
13 choose 3286
14 choose 3364
15 choose 3455
16 choose 3560
17 choose 3680
18 choose 3816
19 choose 3969
20 choose 31,140
25 choose 32,300
40 choose 39,880
100 choose 3161,700

FAQ

What is 17 choose 3?

17 choose 3 is 680

Why is 17 choose 3 the same as 17 choose 14?

Every way to choose 3 of 17 items leaves exactly 14 behind, so choosing 3 and choosing which 14 to leave out are the same count: 17 choose 3 = 17 choose 14 = 680. In the formula n! / (k! (n − k)!) the two factorials below just swap places.

How many ways are there if order matters?

P(17, 3) = 4,080. Each of the 680 combinations can be put in order in 3! = 6 ways, so permutations = combinations × k!: P(n, k) = n! / (n − k)!.

See Also