17 Choose 13

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

What is 17 choose 13?

Answer

17 choose 13 is
2,380

Order doesn't matter, no repeats: 2,380 ways to choose 13 of 17 items.

Choosing 13 of 17 is the same as leaving 4 out: C(17, 13) = C(17, 4).

C(17, 13) = 17! / (13! · 4!)

= C(17, 4)

= (17 × 16 × 15 × 14) / (4 × 3 × 2 × 1)
= 57,120 / 24

= 2,380

If order matters (permutations): P(17, 13) = 17 × 16 × 15 × … × 5 = 14,820,309,504,000 — each combination in 13! = 6,227,020,800 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 13 for other n

n choose 13Answer
13 choose 131
14 choose 1314
15 choose 13105
16 choose 13560
17 choose 132,380
18 choose 138,568
19 choose 1327,132
20 choose 1377,520

FAQ

What is 17 choose 13?

17 choose 13 is 2,380

Why is 17 choose 13 the same as 17 choose 4?

Every way to choose 13 of 17 items leaves exactly 4 behind, so choosing 13 and choosing which 4 to leave out are the same count: 17 choose 13 = 17 choose 4 = 2,380. In the formula n! / (k! (n − k)!) the two factorials below just swap places.

How many ways are there if order matters?

P(17, 13) = 14,820,309,504,000. Each of the 2,380 combinations can be put in order in 13! = 6,227,020,800 ways, so permutations = combinations × k!: P(n, k) = n! / (n − k)!.

See Also