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    20 Choose 17

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

    What is 20 choose 17?

    Answer

    20 choose 17 is
    1,140

    Order doesn't matter, no repeats: 1,140 ways to choose 17 of 20 items.

    Choosing 17 of 20 is the same as leaving 3 out: C(20, 17) = C(20, 3).

    C(20, 17) = 20! / (17! · 3!)

    = C(20, 3)

    = (20 × 19 × 18) / (3 × 2 × 1)
    = 6,840 / 6

    = 1,140

    If order matters (permutations): P(20, 17) = 20 × 19 × 18 × … × 4 = 405,483,668,029,440,000 — each combination in 17! = 355,687,428,096,000 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--17Copy
    <a href="https://calculat.io/en/number/combinations/20--17">20 Choose 17: Combinations C(20, 17) - Calculatio</a>Copy
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    20 Choose 17: Combinations C(20, 17). 20 choose 17 = 1,140: the number of ways to choose 17 of 20 items when order doesn't matter and nothing repeats. C(20, 17) = (20 × 19 × 18) / (3 × 2 × 1).
    https://calculat.io/en/number/combinations/20--17/generated.jpgCopy

    n choose 17 for other n

    n choose 17Answer
    17 choose 171
    18 choose 1718
    19 choose 17171
    20 choose 171,140
    Recalculate with different values ↑

    FAQ

    What is 20 choose 17?

    20 choose 17 is 1,140

    Why is 20 choose 17 the same as 20 choose 3?

    Every way to choose 17 of 20 items leaves exactly 3 behind, so choosing 17 and choosing which 3 to leave out are the same count: 20 choose 17 = 20 choose 3 = 1,140. In the formula n! / (k! (n − k)!) the two factorials below just swap places.

    How many ways are there if order matters?

    P(20, 17) = 405,483,668,029,440,000. Each of the 1,140 combinations can be put in order in 17! = 355,687,428,096,000 ways, so permutations = combinations × k!: P(n, k) = n! / (n − k)!.

    See Also