# Binomial coefficients #

This file defines binomial coefficients and proves simple lemmas (i.e. those not requiring more imports).

## Main definition and results #

• Nat.choose: binomial coefficients, defined inductively
• Nat.choose_eq_factorial_div_factorial: a proof that choose n k = n! / (k! * (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
• Nat.descFactorial_eq_factorial_mul_choose: Relates binomial coefficients to the descending factorial. This is used to prove Nat.choose_le_pow and variants. We provide similar statements for the ascending factorial.
• Nat.multichoose: whereas choose counts combinations, multichoose counts multicombinations. The fact that this is indeed the correct counting function for multisets is proved in Sym.card_sym_eq_multichoose in Data.Sym.Card.
• Nat.multichoose_eq : a proof that multichoose n k = (n + k - 1).choose k. This is central to the "stars and bars" technique in informal mathematics, where we switch between counting multisets of size k over an alphabet of size n to counting strings of k elements ("stars") separated by n-1 dividers ("bars"). See Data.Sym.Card for more detail.

## Tags #

binomial coefficient, combination, multicombination, stars and bars

def Nat.choose :

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

Equations
• x.choose 0 = 1
• Nat.choose 0 n.succ = 0
• n.succ.choose k.succ = n.choose k + n.choose (k + 1)
Instances For
@[simp]
theorem Nat.choose_zero_right (n : ) :
n.choose 0 = 1
@[simp]
theorem Nat.choose_zero_succ (k : ) :
Nat.choose 0 k.succ = 0
theorem Nat.choose_succ_succ (n : ) (k : ) :
n.succ.choose k.succ = n.choose k + n.choose k.succ
theorem Nat.choose_succ_succ' (n : ) (k : ) :
(n + 1).choose (k + 1) = n.choose k + n.choose (k + 1)
theorem Nat.choose_eq_zero_of_lt {n : } {k : } :
n < kn.choose k = 0
@[simp]
theorem Nat.choose_self (n : ) :
n.choose n = 1
@[simp]
theorem Nat.choose_succ_self (n : ) :
n.choose n.succ = 0
@[simp]
theorem Nat.choose_one_right (n : ) :
n.choose 1 = n
theorem Nat.triangle_succ (n : ) :
(n + 1) * (n + 1 - 1) / 2 = n * (n - 1) / 2 + n
theorem Nat.choose_two_right (n : ) :
n.choose 2 = n * (n - 1) / 2

choose n 2 is the n-th triangle number.

theorem Nat.choose_pos {n : } {k : } :
k n0 < n.choose k
theorem Nat.choose_eq_zero_iff {n : } {k : } :
n.choose k = 0 n < k
theorem Nat.succ_mul_choose_eq (n : ) (k : ) :
n.succ * n.choose k = n.succ.choose k.succ * k.succ
theorem Nat.choose_mul_factorial_mul_factorial {n : } {k : } :
k nn.choose k * k.factorial * (n - k).factorial = n.factorial
theorem Nat.choose_mul {n : } {k : } {s : } (hkn : k n) (hsk : s k) :
n.choose k * k.choose s = n.choose s * (n - s).choose (k - s)
theorem Nat.choose_eq_factorial_div_factorial {n : } {k : } (hk : k n) :
n.choose k = n.factorial / (k.factorial * (n - k).factorial)
theorem Nat.add_choose (i : ) (j : ) :
(i + j).choose j = (i + j).factorial / (i.factorial * j.factorial)
theorem Nat.add_choose_mul_factorial_mul_factorial (i : ) (j : ) :
(i + j).choose j * i.factorial * j.factorial = (i + j).factorial
theorem Nat.factorial_mul_factorial_dvd_factorial {n : } {k : } (hk : k n) :
k.factorial * (n - k).factorial n.factorial
theorem Nat.factorial_mul_factorial_dvd_factorial_add (i : ) (j : ) :
i.factorial * j.factorial (i + j).factorial
@[simp]
theorem Nat.choose_symm {n : } {k : } (hk : k n) :
n.choose (n - k) = n.choose k
theorem Nat.choose_symm_of_eq_add {n : } {a : } {b : } (h : n = a + b) :
n.choose a = n.choose b
theorem Nat.choose_symm_add {a : } {b : } :
(a + b).choose a = (a + b).choose b
theorem Nat.choose_symm_half (m : ) :
(2 * m + 1).choose (m + 1) = (2 * m + 1).choose m
theorem Nat.choose_succ_right_eq (n : ) (k : ) :
n.choose (k + 1) * (k + 1) = n.choose k * (n - k)
@[simp]
theorem Nat.choose_succ_self_right (n : ) :
(n + 1).choose n = n + 1
theorem Nat.choose_mul_succ_eq (n : ) (k : ) :
n.choose k * (n + 1) = (n + 1).choose k * (n + 1 - k)
theorem Nat.ascFactorial_eq_factorial_mul_choose (n : ) (k : ) :
(n + 1).ascFactorial k = k.factorial * (n + k).choose k
theorem Nat.ascFactorial_eq_factorial_mul_choose' (n : ) (k : ) :
n.ascFactorial k = k.factorial * (n + k - 1).choose k
theorem Nat.factorial_dvd_ascFactorial (n : ) (k : ) :
k.factorial n.ascFactorial k
theorem Nat.choose_eq_asc_factorial_div_factorial (n : ) (k : ) :
(n + k).choose k = (n + 1).ascFactorial k / k.factorial
theorem Nat.choose_eq_asc_factorial_div_factorial' (n : ) (k : ) :
(n + k - 1).choose k = n.ascFactorial k / k.factorial
theorem Nat.descFactorial_eq_factorial_mul_choose (n : ) (k : ) :
n.descFactorial k = k.factorial * n.choose k
theorem Nat.factorial_dvd_descFactorial (n : ) (k : ) :
k.factorial n.descFactorial k
theorem Nat.choose_eq_descFactorial_div_factorial (n : ) (k : ) :
n.choose k = n.descFactorial k / k.factorial
def Nat.fast_choose (n : ) (k : ) :

A faster implementation of choose, to be used during bytecode evaluation and in compiled code.

Equations
• n.fast_choose k = n.descFactorial k / k.factorial
Instances For
@[csimp]

### Inequalities #

theorem Nat.choose_le_succ_of_lt_half_left {r : } {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.choose r n.choose (n / 2)

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

#### Inequalities about increasing the first argument #

theorem Nat.choose_le_succ (a : ) (c : ) :
a.choose c a.succ.choose c
theorem Nat.choose_le_add (a : ) (b : ) (c : ) :
a.choose c (a + b).choose c
theorem Nat.choose_le_choose {a : } {b : } (c : ) (h : a b) :
a.choose c b.choose c
theorem Nat.choose_mono (b : ) :
Monotone fun (a : ) => a.choose b

#### Multichoose #

Whereas choose n k is the number of subsets of cardinality k from a type of cardinality n, multichoose n k is the number of multisets of cardinality k from a type of cardinality n.

Alternatively, whereas choose n k counts the number of combinations, i.e. ways to select k items (up to permutation) from n items without replacement, multichoose n k counts the number of multicombinations, i.e. ways to select k items (up to permutation) from n items with replacement.

Note that multichoose is not the multinomial coefficient, although it can be computed in terms of multinomial coefficients. For details see https://mathworld.wolfram.com/Multichoose.html

TODO: Prove that choose (-n) k = (-1)^k * multichoose n k, where choose is the generalized binomial coefficient.

def Nat.multichoose :

multichoose n k is the number of multisets of cardinality k from a type of cardinality n.

Equations
• x.multichoose 0 = 1
• Nat.multichoose 0 n.succ = 0
• n.succ.multichoose k.succ = n.multichoose (k + 1) + (n + 1).multichoose k
Instances For
@[simp]
theorem Nat.multichoose_zero_right (n : ) :
n.multichoose 0 = 1
@[simp]
theorem Nat.multichoose_succ_succ (n : ) (k : ) :
(n + 1).multichoose (k + 1) = n.multichoose (k + 1) + (n + 1).multichoose k
@[simp]
theorem Nat.multichoose_one (k : ) :
= 1
@[simp]
theorem Nat.multichoose_two (k : ) :
= k + 1
@[simp]
theorem Nat.multichoose_one_right (n : ) :
n.multichoose 1 = n
theorem Nat.multichoose_eq (n : ) (k : ) :
n.multichoose k = (n + k - 1).choose k