16 Choose 7

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

What is 16 choose 7?

Answer

16 choose 7 is
11,440

Order doesn't matter, no repeats: 11,440 ways to choose 7 of 16 items.

C(16, 7) = 16! / (7! · 9!)

= (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, 7) = 16 × 15 × 14 × … × 10 = 57,657,600 — each combination in 7! = 5,040 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 7 for other n

n choose 7Answer
7 choose 71
8 choose 78
9 choose 736
10 choose 7120
11 choose 7330
12 choose 7792
13 choose 71,716
14 choose 73,432
15 choose 76,435
16 choose 711,440
17 choose 719,448
18 choose 731,824
19 choose 750,388
20 choose 777,520

FAQ

What is 16 choose 7?

16 choose 7 is 11,440

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

Every way to choose 7 of 16 items leaves exactly 9 behind, so choosing 7 and choosing which 9 to leave out are the same count: 16 choose 7 = 16 choose 9 = 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, 7) = 57,657,600. Each of the 11,440 combinations can be put in order in 7! = 5,040 ways, so permutations = combinations × k!: P(n, k) = n! / (n − k)!.

See Also