17 Choose 14

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

What is 17 choose 14?

Answer

17 choose 14 is
680

Order doesn't matter, no repeats: 680 ways to choose 14 of 17 items.

Choosing 14 of 17 is the same as leaving 3 out: C(17, 14) = C(17, 3).

C(17, 14) = 17! / (14! · 3!)

= C(17, 3)

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

= 680

If order matters (permutations): P(17, 14) = 17 × 16 × 15 × … × 4 = 59,281,238,016,000 — each combination in 14! = 87,178,291,200 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 14 for other n

n choose 14Answer
14 choose 141
15 choose 1415
16 choose 14120
17 choose 14680
18 choose 143,060
19 choose 1411,628
20 choose 1438,760

FAQ

What is 17 choose 14?

17 choose 14 is 680

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

Every way to choose 14 of 17 items leaves exactly 3 behind, so choosing 14 and choosing which 3 to leave out are the same count: 17 choose 14 = 17 choose 3 = 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, 14) = 59,281,238,016,000. Each of the 680 combinations can be put in order in 14! = 87,178,291,200 ways, so permutations = combinations × k!: P(n, k) = n! / (n − k)!.

See Also