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    20 Choose 7

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

    What is 20 choose 7?

    Answer

    20 choose 7 is
    77,520

    Order doesn't matter, no repeats: 77,520 ways to choose 7 of 20 items.

    C(20, 7) = 20! / (7! · 13!)

    = (20 × 19 × 18 × … × 14) / (7 × 6 × 5 × … × 1)

    7 factors on top and bottom

    = 390,700,800 / 5,040

    = 77,520

    If order matters (permutations): P(20, 7) = 20 × 19 × 18 × … × 14 = 390,700,800 — each combination in 7! = 5,040 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.

    See Also

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    https://calculat.io/en/number/combinations/20--7Copy
    <a href="https://calculat.io/en/number/combinations/20--7">20 Choose 7: Combinations C(20, 7) - Calculatio</a>Copy
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    20 Choose 7: Combinations C(20, 7). 20 choose 7 = 77,520: the number of ways to choose 7 of 20 items when order doesn't matter and nothing repeats.
    https://calculat.io/en/number/combinations/20--7/generated.jpgCopy

    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
    Recalculate with different values ↑

    FAQ

    What is 20 choose 7?

    20 choose 7 is 77,520

    Why is 20 choose 7 the same as 20 choose 13?

    Every way to choose 7 of 20 items leaves exactly 13 behind, so choosing 7 and choosing which 13 to leave out are the same count: 20 choose 7 = 20 choose 13 = 77,520. In the formula n! / (k! (n − k)!) the two factorials below just swap places.

    How many ways are there if order matters?

    P(20, 7) = 390,700,800. Each of the 77,520 combinations can be put in order in 7! = 5,040 ways, so permutations = combinations × k!: P(n, k) = n! / (n − k)!.

    See Also