20 Choose 7

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

What is 20 choose 7?

Answer

20 choose 7 is
77,520

Order doesn't matter, no repeats: 77,520 ways to choose 7 of 20 items.

C(20, 7) = 20! / (7! · 13!)

= (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, 7) = 20 × 19 × 18 × … × 14 = 390,700,800 — each combination in 7! = 5,040 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 7 for other n

n choose 7Answer
7 choose 71
8 choose 78
9 choose 736
10 choose 7120
11 choose 7330
12 choose 7792
13 choose 71,716
14 choose 73,432
15 choose 76,435
16 choose 711,440
17 choose 719,448
18 choose 731,824
19 choose 750,388
20 choose 777,520

FAQ

What is 20 choose 7?

20 choose 7 is 77,520

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

Every way to choose 7 of 20 items leaves exactly 13 behind, so choosing 7 and choosing which 13 to leave out are the same count: 20 choose 7 = 20 choose 13 = 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, 7) = 390,700,800. Each of the 77,520 combinations can be put in order in 7! = 5,040 ways, so permutations = combinations × k!: P(n, k) = n! / (n − k)!.

See Also