16 Choose 8

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

What is 16 choose 8?

Answer

16 choose 8 is
12,870

Order doesn't matter, no repeats: 12,870 ways to choose 8 of 16 items.

C(16, 8) = 16! / (8! · 8!)

= (16 × 15 × 14 × … × 9) / (8 × 7 × 6 × … × 1)

8 factors on top and bottom

= 518,918,400 / 40,320

= 12,870

If order matters (permutations): P(16, 8) = 16 × 15 × 14 × … × 9 = 518,918,400 — each combination in 8! = 40,320 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 8 for other n

n choose 8Answer
8 choose 81
9 choose 89
10 choose 845
11 choose 8165
12 choose 8495
13 choose 81,287
14 choose 83,003
15 choose 86,435
16 choose 812,870
17 choose 824,310
18 choose 843,758
19 choose 875,582
20 choose 8125,970

FAQ

What is 16 choose 8?

16 choose 8 is 12,870

Why is 16 choose 8 the largest number in row 16?

In row 16 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 = 8, so 16 choose 8 = 12,870 is the largest: 16 choose 7 is only 11,440.

How many ways are there if order matters?

P(16, 8) = 518,918,400. Each of the 12,870 combinations can be put in order in 8! = 40,320 ways, so permutations = combinations × k!: P(n, k) = n! / (n − k)!.

See Also