20 Choose 10

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

What is 20 choose 10?

Answer

20 choose 10 is
184,756

Order doesn't matter, no repeats: 184,756 ways to choose 10 of 20 items.

C(20, 10) = 20! / (10! · 10!)

= (20 × 19 × 18 × … × 11) / (10 × 9 × 8 × … × 1)

10 factors on top and bottom

= 670,442,572,800 / 3,628,800

= 184,756

If order matters (permutations): P(20, 10) = 20 × 19 × 18 × … × 11 = 670,442,572,800 — each combination in 10! = 3,628,800 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 10 for other n

n choose 10Answer
2 choose 100
10 choose 101
11 choose 1011
12 choose 1066
13 choose 10286
14 choose 101,001
15 choose 103,003
16 choose 108,008
17 choose 1019,448
18 choose 1043,758
19 choose 1092,378
20 choose 10184,756
100 choose 1017,310,309,456,440

FAQ

What is 20 choose 10?

20 choose 10 is 184,756

Why is 20 choose 10 the largest number in row 20?

In row 20 of Pascal's triangle the numbers grow towards the middle and shrink after it, because n choose k = n choose (n − k). With an even n the middle is k = 10, so 20 choose 10 = 184,756 is the largest: 20 choose 9 is only 167,960.

How many ways are there if order matters?

P(20, 10) = 670,442,572,800. Each of the 184,756 combinations can be put in order in 10! = 3,628,800 ways, so permutations = combinations × k!: P(n, k) = n! / (n − k)!.

See Also