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    10 Choose 5

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

    What is 10 choose 5?

    Answer

    10 choose 5 is
    252

    Order doesn't matter, no repeats: 252 ways to choose 5 of 10 items.

    C(10, 5) = 10! / (5! · 5!)

    = (10 × 9 × 8 × 7 × 6) / (5 × 4 × 3 × 2 × 1)
    = 30,240 / 120

    = 252

    If order matters (permutations): P(10, 5) = 10 × 9 × 8 × 7 × 6 = 30,240 — each combination in 5! = 120 orders.

    Row 10 of Pascal's triangle
    k10 choose k
    01
    110
    245
    3120
    4210
    5252
    6210
    7120
    845
    910
    101

    Row sum: 210 = 1,024 — all subsets of 10 items.

    See Also

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    https://calculat.io/en/number/combinations/10--5Copy
    <a href="https://calculat.io/en/number/combinations/10--5">10 Choose 5: Combinations C(10, 5) - Calculatio</a>Copy
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    10 Choose 5: Combinations C(10, 5). 10 choose 5 = 252: the number of ways to choose 5 of 10 items when order doesn't matter and nothing repeats.
    https://calculat.io/en/number/combinations/10--5/generated.jpgCopy

    n choose 5 for other n

    n choose 5Answer
    2 choose 50
    5 choose 51
    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
    Recalculate with different values ↑

    FAQ

    What is 10 choose 5?

    10 choose 5 is 252

    Why is 10 choose 5 the largest number in row 10?

    In row 10 of Pascal's triangle the numbers grow towards the middle and shrink after it, because n choose k = n choose (n − k). With an even n the middle is k = 5, so 10 choose 5 = 252 is the largest: 10 choose 4 is only 210.

    How many ways are there if order matters?

    P(10, 5) = 30,240. Each of the 252 combinations can be put in order in 5! = 120 ways, so permutations = combinations × k!: P(n, k) = n! / (n − k)!.

    See Also