MathTools

Combinatorics

Factorials, permutations (nPr) and combinations (nCr), worked out exactly with big integers — no overflow, no rounding.
Computed
The result and the worked steps are shown below.
C(5, 2) =

10

Step-by-step solution

  1. 1

    Definition

    C(n, k) = n! / (k! · (n − k)!) counts unordered selections of k items from n. Here C(5, 2) = 5! / (2! · 3!).

    (52)=5!2! (5−2)!=5!2! 3!\binom{5}{2} = \frac{5!}{2!\,(5 - 2)!} = \frac{5!}{2!\,3!}
  2. 2

    Cancel the factorial tail

    5! / 3! = 5 × 4 (the surviving numerator), and k! = 1 × 2 = 2.

  3. 3

    Substitute back

    C(5, 2) = (5 × 4) / (1 × 2) = 20 / 2.

    (52)=5×41×2=202\binom{5}{2} = \frac{5 \times 4}{1 \times 2} = \frac{20}{2}
  4. 4

    Divide

    20 / 2 = 10

  5. 5

    Result

    C(5, 2) = 10

    (52)=10\binom{5}{2} = 10