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    16 Choose 3

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

    What is 16 choose 3?

    Answer

    16 choose 3 is
    560

    Order doesn't matter, no repeats: 560 ways to choose 3 of 16 items.

    C(16, 3) = 16! / (3! · 13!)

    = (16 × 15 × 14) / (3 × 2 × 1)
    = 3,360 / 6

    = 560

    If order matters (permutations): P(16, 3) = 16 × 15 × 14 = 3,360 — each combination in 3! = 6 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.

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    https://calculat.io/en/number/combinations/16--3Copy
    <a href="https://calculat.io/en/number/combinations/16--3">16 Choose 3: Combinations C(16, 3) - Calculatio</a>Copy
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    16 Choose 3: Combinations C(16, 3). 16 choose 3 = 560: the number of ways to choose 3 of 16 items when order doesn't matter and nothing repeats. C(16, 3) = (16 × 15 × 14) / (3 × 2 × 1).
    https://calculat.io/en/number/combinations/16--3/generated.jpgCopy

    n choose 3 for other n

    n choose 3Answer
    9 choose 384
    10 choose 3120
    11 choose 3165
    12 choose 3220
    13 choose 3286
    14 choose 3364
    15 choose 3455
    16 choose 3560
    17 choose 3680
    18 choose 3816
    19 choose 3969
    20 choose 31,140
    25 choose 32,300
    40 choose 39,880
    100 choose 3161,700
    Recalculate with different values ↑

    FAQ

    What is 16 choose 3?

    16 choose 3 is 560

    Why is 16 choose 3 the same as 16 choose 13?

    Every way to choose 3 of 16 items leaves exactly 13 behind, so choosing 3 and choosing which 13 to leave out are the same count: 16 choose 3 = 16 choose 13 = 560. In the formula n! / (k! (n − k)!) the two factorials below just swap places.

    How many ways are there if order matters?

    P(16, 3) = 3,360. Each of the 560 combinations can be put in order in 3! = 6 ways, so permutations = combinations × k!: P(n, k) = n! / (n − k)!.

    See Also