BINOMIAL COEFFICIENT

Binomial coefficient: n, k ∈ ℕ | n ≥ k ≥ 0; C(n, k) = n!/k!(n-k)!

The binomial coefficient expresses the number of unordered subset of k elements from a fixed set of n elements.

C(3, 2) = 3!/2!(3-2)! = 6/2 = 3

C(5, 2) = 5!/2!(5-2)! = 120/12 = 10

The Binomial Coefficient C(n, k) represents the number of subsets of k objects of the n given objects.

C(3, 2) = 3!/2!(3-2)! = 6/2 = 3; {a, b, c}: {a, b}, {a, c}, {b, c}; For 3 objects there are 3 subsets of 2 objects.

C(90, 6) = 90!/6!(90-6)! = 90!/6!⋅84! = 622.614.630; 1/C(90, 6) = 10^{-9}

Binomial Coefficient Properties: C(n, k) ∈ ℕ, ∀ n, k ∈ ℕ; C(n, 0) = C(n, n) = 1; C(n, k) = C(n, n-k); C(n, k) = C(n-1, k) + (n-1, k-1)

1

1, 1

1, 2, 1

1, 3, 3, 1

1, 4, 6, 4, 1

1, 5, 10, 10, 5, 1

1, 6, 15, 20, 15, 6, 1

1, 7, 21, 35, 35, 21, 7, 1

1, 8, 28, 56, 70, 56, 28, 8, 1