20 Choose 15

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

What is 20 choose 15?

Answer

20 choose 15 is
15,504

Order doesn't matter, no repeats: 15,504 ways to choose 15 of 20 items.

Choosing 15 of 20 is the same as leaving 5 out: C(20, 15) = C(20, 5).

C(20, 15) = 20! / (15! · 5!)

= C(20, 5)

= (20 × 19 × 18 × 17 × 16) / (5 × 4 × 3 × 2 × 1)
= 1,860,480 / 120

= 15,504

If order matters (permutations): P(20, 15) = 20 × 19 × 18 × … × 6 = 20,274,183,401,472,000 — each combination in 15! = 1,307,674,368,000 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 15 for other n

n choose 15Answer
15 choose 151
16 choose 1516
17 choose 15136
18 choose 15816
19 choose 153,876
20 choose 1515,504

FAQ

What is 20 choose 15?

20 choose 15 is 15,504

Why is 20 choose 15 the same as 20 choose 5?

Every way to choose 15 of 20 items leaves exactly 5 behind, so choosing 15 and choosing which 5 to leave out are the same count: 20 choose 15 = 20 choose 5 = 15,504. In the formula n! / (k! (n − k)!) the two factorials below just swap places.

How many ways are there if order matters?

P(20, 15) = 20,274,183,401,472,000. Each of the 15,504 combinations can be put in order in 15! = 1,307,674,368,000 ways, so permutations = combinations × k!: P(n, k) = n! / (n − k)!.

See Also