18 Choose 9

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

What is 18 choose 9?

Answer

18 choose 9 is
48,620

Order doesn't matter, no repeats: 48,620 ways to choose 9 of 18 items.

C(18, 9) = 18! / (9! · 9!)

= (18 × 17 × 16 × … × 10) / (9 × 8 × 7 × … × 1)

9 factors on top and bottom

= 17,643,225,600 / 362,880

= 48,620

If order matters (permutations): P(18, 9) = 18 × 17 × 16 × … × 10 = 17,643,225,600 — each combination in 9! = 362,880 orders.

Row 18 of Pascal's triangle
k18 choose k
01
118
2153
3816
43,060
58,568
618,564
731,824
843,758
948,620
1043,758
1131,824
1218,564
138,568
143,060
15816
16153
1718
181

Row sum: 218 = 262,144 — all subsets of 18 items.

See Also

Recalculate with different values ↑

n choose 9 for other n

n choose 9Answer
9 choose 91
10 choose 910
11 choose 955
12 choose 9220
13 choose 9715
14 choose 92,002
15 choose 95,005
16 choose 911,440
17 choose 924,310
18 choose 948,620
19 choose 992,378
20 choose 9167,960

FAQ

What is 18 choose 9?

18 choose 9 is 48,620

Why is 18 choose 9 the largest number in row 18?

In row 18 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 = 9, so 18 choose 9 = 48,620 is the largest: 18 choose 8 is only 43,758.

How many ways are there if order matters?

P(18, 9) = 17,643,225,600. Each of the 48,620 combinations can be put in order in 9! = 362,880 ways, so permutations = combinations × k!: P(n, k) = n! / (n − k)!.

See Also