20 Choose 13

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

What is 20 choose 13?

Answer

20 choose 13 is
77,520

Order doesn't matter, no repeats: 77,520 ways to choose 13 of 20 items.

Choosing 13 of 20 is the same as leaving 7 out: C(20, 13) = C(20, 7).

C(20, 13) = 20! / (13! · 7!)

= C(20, 7)

= (20 × 19 × 18 × … × 14) / (7 × 6 × 5 × … × 1)

7 factors on top and bottom

= 390,700,800 / 5,040

= 77,520

If order matters (permutations): P(20, 13) = 20 × 19 × 18 × … × 8 = 482,718,652,416,000 — each combination in 13! = 6,227,020,800 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 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 20 choose 13?

20 choose 13 is 77,520

Why is 20 choose 13 the same as 20 choose 7?

Every way to choose 13 of 20 items leaves exactly 7 behind, so choosing 13 and choosing which 7 to leave out are the same count: 20 choose 13 = 20 choose 7 = 77,520. In the formula n! / (k! (n − k)!) the two factorials below just swap places.

How many ways are there if order matters?

P(20, 13) = 482,718,652,416,000. Each of the 77,520 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