mathlib documentation

data.nat.choose.basic

Binomial coefficients

This file contains a definition of binomial coefficients and simple lemmas (i.e. those not requiring more imports).

Main definition and results

nat.choose: binomial coefficients, defined inductively
nat.choose_eq_fact_div_fact: a proof that choose n k = fact n / (fact k * fact (n - k))
nat.choose_symm: symmetry of binomial coefficients
nat.choose_le_succ_of_lt_half_left: choose n k is increasing for small values of k
nat.choose_le_middle: choose n r is maximised when r is n/2

def nat.choose (a a_1 : ℕ) :

choose n k is the number of k-element subsets in an n-element set. Also known as binomial coefficients.

Equations

(n + 1).choose (k + 1) = n.choose k + n.choose (k + 1)
n.succ.choose 0 = 1
0.choose (k + 1) = 0
0.choose 0 = 1

@[simp]

theorem nat.choose_zero_right (n : ℕ) :

n.choose 0 = 1

@[simp]

theorem nat.choose_zero_succ (k : ℕ) :

0.choose k.succ = 0

theorem nat.choose_succ_succ (n k : ℕ) :

n.succ.choose k.succ = n.choose k + n.choose k.succ

theorem nat.choose_eq_zero_of_lt {n k : ℕ} (a : n < k) :

n.choose k = 0

@[simp]

theorem nat.choose_self (n : ℕ) :

n.choose n = 1

@[simp]

theorem nat.choose_succ_self (n : ℕ) :

n.choose n.succ = 0

@[simp]

theorem nat.choose_one_right (n : ℕ) :

n.choose 1 = n

theorem nat.triangle_succ (n : ℕ) :

(n + 1) * (n + 1 - 1) / 2 = n * (n - 1) / 2 + n

theorem nat.choose_two_right (n : ℕ) :

n.choose 2 = n * (n - 1) / 2

choose n 2 is the n-th triangle number.

theorem nat.choose_pos {n k : ℕ} (a : k ≤ n) :

0 < n.choose k

theorem nat.succ_mul_choose_eq (n k : ℕ) :

(n.succ) * n.choose k = (n.succ.choose k.succ) * k.succ

theorem nat.choose_mul_fact_mul_fact {n k : ℕ} (a : k ≤ n) :

((n.choose k) * k.fact) * (n - k).fact = n.fact

theorem nat.choose_eq_fact_div_fact {n k : ℕ} (hk : k ≤ n) :

n.choose k = n.fact / (k.fact) * (n - k).fact

theorem nat.fact_mul_fact_dvd_fact {n k : ℕ} (hk : k ≤ n) :

(k.fact) * (n - k).fact ∣ n.fact

@[simp]

theorem nat.choose_symm {n k : ℕ} (hk : k ≤ n) :

n.choose (n - k) = n.choose k

theorem nat.choose_symm_of_eq_add {n a b : ℕ} (h : n = a + b) :

n.choose a = n.choose b

theorem nat.choose_symm_add {a b : ℕ} :

(a + b).choose a = (a + b).choose b

theorem nat.choose_symm_half (m : ℕ) :

(2 * m + 1).choose (m + 1) = (2 * m + 1).choose m

theorem nat.choose_succ_right_eq (n k : ℕ) :

(n.choose (k + 1)) * (k + 1) = (n.choose k) * (n - k)

@[simp]

theorem nat.choose_succ_self_right (n : ℕ) :

(n + 1).choose n = n + 1

theorem nat.choose_mul_succ_eq (n k : ℕ) :

(n.choose k) * (n + 1) = ((n + 1).choose k) * (n + 1 - k)

Inequalities

theorem nat.choose_le_succ_of_lt_half_left {r n : ℕ} (h : r < n / 2) :

n.choose r ≤ n.choose (r + 1)

Show that nat.choose is increasing for small values of the right argument.

theorem nat.choose_le_middle (r n : ℕ) :

n.choose r ≤ n.choose (n / 2)

choose n r is maximised when r is n/2.