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    20 Choose 5

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

    What is 20 choose 5?

    Answer

    20 choose 5 is
    15,504

    Order doesn't matter, no repeats: 15,504 ways to choose 5 of 20 items.

    C(20, 5) = 20! / (5! · 15!)

    = (20 × 19 × 18 × 17 × 16) / (5 × 4 × 3 × 2 × 1)
    = 1,860,480 / 120

    = 15,504

    If order matters (permutations): P(20, 5) = 20 × 19 × 18 × 17 × 16 = 1,860,480 — each combination in 5! = 120 orders.

    Row 20 of Pascal's triangle
    k20 choose k
    01
    120
    2190
    31,140
    44,845
    515,504
    638,760
    777,520
    8125,970
    9167,960
    10184,756
    11167,960
    12125,970
    1377,520
    1438,760
    1515,504
    164,845
    171,140
    18190
    1920
    201

    Row sum: 220 = 1,048,576 — all subsets of 20 items.

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    https://calculat.io/en/number/combinations/20--5Copy
    <a href="https://calculat.io/en/number/combinations/20--5">20 Choose 5: Combinations C(20, 5) - Calculatio</a>Copy
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    20 Choose 5: Combinations C(20, 5). 20 choose 5 = 15,504: the number of ways to choose 5 of 20 items when order doesn't matter and nothing repeats.
    https://calculat.io/en/number/combinations/20--5/generated.jpgCopy

    n choose 5 for other n

    n choose 5Answer
    6 choose 56
    7 choose 521
    8 choose 556
    9 choose 5126
    10 choose 5252
    11 choose 5462
    12 choose 5792
    13 choose 51,287
    14 choose 52,002
    15 choose 53,003
    16 choose 54,368
    17 choose 56,188
    18 choose 58,568
    19 choose 511,628
    20 choose 515,504
    Recalculate with different values ↑

    FAQ

    What is 20 choose 5?

    20 choose 5 is 15,504

    Why is 20 choose 5 the same as 20 choose 15?

    Every way to choose 5 of 20 items leaves exactly 15 behind, so choosing 5 and choosing which 15 to leave out are the same count: 20 choose 5 = 20 choose 15 = 15,504. In the formula n! / (k! (n − k)!) the two factorials below just swap places.

    How many ways are there if order matters?

    P(20, 5) = 1,860,480. Each of the 15,504 combinations can be put in order in 5! = 120 ways, so permutations = combinations × k!: P(n, k) = n! / (n − k)!.

    See Also