17 Choose 16

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

What is 17 choose 16?

Answer

17 choose 16 is
17

Order doesn't matter, no repeats: 17 ways to choose 16 of 17 items.

Choosing 16 of 17 is the same as leaving 1 out: C(17, 16) = C(17, 1).

C(17, 16) = 17! / (16! · 1!)

= C(17, 1)

= 17 / 1

= 17

If order matters (permutations): P(17, 16) = 17 × 16 × 15 × … × 2 = 355,687,428,096,000 — each combination in 16! = 20,922,789,888,000 orders.

Row 17 of Pascal's triangle
k17 choose k
01
117
2136
3680
42,380
56,188
612,376
719,448
824,310
924,310
1019,448
1112,376
126,188
132,380
14680
15136
1617
171

Row sum: 217 = 131,072 — all subsets of 17 items.

See Also

Recalculate with different values ↑

n choose 16 for other n

n choose 16Answer
16 choose 161
17 choose 1617
18 choose 16153
19 choose 16969
20 choose 164,845

FAQ

What is 17 choose 16?

17 choose 16 is 17

Why is 17 choose 16 the same as 17 choose 1?

Every way to choose 16 of 17 items leaves exactly 1 behind, so choosing 16 and choosing which one to leave out are the same count: 17 choose 16 = 17 choose 1 = 17. In the formula n! / (k! (n − k)!) the two factorials below just swap places.

How many ways are there if order matters?

P(17, 16) = 355,687,428,096,000. Each of the 17 combinations can be put in order in 16! = 20,922,789,888,000 ways, so permutations = combinations × k!: P(n, k) = n! / (n − k)!.

See Also