20 Choose 17

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

What is 20 choose 17?

Answer

20 choose 17 is
1,140

Order doesn't matter, no repeats: 1,140 ways to choose 17 of 20 items.

Choosing 17 of 20 is the same as leaving 3 out: C(20, 17) = C(20, 3).

C(20, 17) = 20! / (17! · 3!)

= C(20, 3)

= (20 × 19 × 18) / (3 × 2 × 1)
= 6,840 / 6

= 1,140

If order matters (permutations): P(20, 17) = 20 × 19 × 18 × … × 4 = 405,483,668,029,440,000 — each combination in 17! = 355,687,428,096,000 orders.

Row 20 of Pascal's triangle
k20 choose k
01
120
2190
31,140
44,845
515,504
638,760
777,520
8125,970
9167,960
10184,756
11167,960
12125,970
1377,520
1438,760
1515,504
164,845
171,140
18190
1920
201

Row sum: 220 = 1,048,576 — all subsets of 20 items.

See Also

Recalculate with different values ↑

n choose 17 for other n

n choose 17Answer
17 choose 171
18 choose 1718
19 choose 17171
20 choose 171,140

FAQ

What is 20 choose 17?

20 choose 17 is 1,140

Why is 20 choose 17 the same as 20 choose 3?

Every way to choose 17 of 20 items leaves exactly 3 behind, so choosing 17 and choosing which 3 to leave out are the same count: 20 choose 17 = 20 choose 3 = 1,140. In the formula n! / (k! (n − k)!) the two factorials below just swap places.

How many ways are there if order matters?

P(20, 17) = 405,483,668,029,440,000. Each of the 1,140 combinations can be put in order in 17! = 355,687,428,096,000 ways, so permutations = combinations × k!: P(n, k) = n! / (n − k)!.

See Also