16 Choose 9

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

What is 16 choose 9?

Answer

16 choose 9 is
11,440

Order doesn't matter, no repeats: 11,440 ways to choose 9 of 16 items.

Choosing 9 of 16 is the same as leaving 7 out: C(16, 9) = C(16, 7).

C(16, 9) = 16! / (9! · 7!)

= C(16, 7)

= (16 × 15 × 14 × … × 10) / (7 × 6 × 5 × … × 1)

7 factors on top and bottom

= 57,657,600 / 5,040

= 11,440

If order matters (permutations): P(16, 9) = 16 × 15 × 14 × … × 8 = 4,151,347,200 — each combination in 9! = 362,880 orders.

Row 16 of Pascal's triangle
k16 choose k
01
116
2120
3560
41,820
54,368
68,008
711,440
812,870
911,440
108,008
114,368
121,820
13560
14120
1516
161

Row sum: 216 = 65,536 — all subsets of 16 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 16 choose 9?

16 choose 9 is 11,440

Why is 16 choose 9 the same as 16 choose 7?

Every way to choose 9 of 16 items leaves exactly 7 behind, so choosing 9 and choosing which 7 to leave out are the same count: 16 choose 9 = 16 choose 7 = 11,440. In the formula n! / (k! (n − k)!) the two factorials below just swap places.

How many ways are there if order matters?

P(16, 9) = 4,151,347,200. Each of the 11,440 combinations can be put in order in 9! = 362,880 ways, so permutations = combinations × k!: P(n, k) = n! / (n − k)!.

See Also