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The beta function is B(a, b) = Γ(a)Γ(b)/Γ(a + b), equal to the integral of x^(a−1)(1−x)^(b−1) from 0 to 1. It is symmetric in its arguments and underpins the beta distribution.
Beta from gamma
B(a, b) = Γ(a)Γ(b) / Γ(a + b)
Whole-number case
B(m, n) = 1 / [(m + n − 1) · C(m + n − 2, m − 1)]
Γ(3)Γ(5)/Γ(8) = 2 × 24 / 5040 = 1/105 ≈ 0.0095238.
Swapping a and b just reflects the integrand about x = ½, which leaves the integral unchanged.