2.5 Combinations of Multisets

Thm 2.5.1 Let S = {∞ · a_{1}, ..., ∞ · a_{k}}. Then the number of r-combinations of S is

Proof: The number of r-combinations of S

= the number of integral solutions to the equation

x_{1} + x_{2} + ... + x_{k} = r (∗)

where x_{i} ≥ 0 ∀i (and where x_{i} = the number of a_{i}’s chosen for an r−combination).

= the number of permutations of {r · 1, (k − 1) · +} by the following:

Suppose (c_{1}, c_{2}, ..., c_{k}) is a solution to (*). This corresponds to the permutation 11...1 + 1..1 + ... + 11..1,

where there are k − 1 +’s and c_{1} 1’s before the first +, c_{i} 1’s between the (i-1)th and ith +’s for i = 2, ..., k − 1, and c_{k} 1’s after the last +. Since c_{1} + c_{2} + ... + c_{k} = r, there are r 1’s, and thus 11...1 + 1..1 + ... + 11..1 is a permutations of {r · 1, (k − 1) · +}.

A permutation of {r · 1, (k − 1) · +} corresponds to a solution (c_{1}, c_{2}, ..., c_{k}) of (*) where c_{1} = the number of 1’s before the first +, c_{i} = the number of 1’s between the (i-1)th and ith +’s for i = 2, ..., k − 1, and c_{k} = the number of 1’s after the last +. Since there are r 1’s, c_{1} + c_{2} + ... + c_{k} = r.

The number of permutations of {r · 1, (k − 1) · +} is Corollary: Let S = {r · a_{1}, ..., r · a_{k}}. Then the number of r-combinations of S is Proof:

Some examples S = {∞ · a_{1}, ∞ · a_{2}, ..., ∞ · a_{5}}. Then a 4-combination of S is {a_{3}, a_{3}, a_{3}, a_{5}} Suppose x_{1} + x_{2} + x_{3} + x_{4} + x_{5} = 4. Then (x_{1}, x_{2}, x_{3}, x_{4}, x_{5}) = (0, 0, 3, 0, 1) is a solution. + + 111 + +1 is a permutation of {4 · 1, (5 − 1) · +} (x_{1}, x_{2}, x_{3}, x_{4}, x_{5}) = (2, 1, 0, 1, 0) is a solution to x_{1} + x_{2} + x_{3} + x_{4} + x_{5} = 4. 11 + 1 + +1+ is a permutation of {4 · 1, (5 − 1) · +} A 4-combination of S is {a_{1}, a_{1}, a_{2}, a_{4}} + + + + 4 is a permutation of {4 · 1, (5 − 1) · +} A 4-combination of S is {a_{5}, a_{5}, a_{5}, a_{5}} (x_{1}, x_{2}, x_{3}, x_{4}, x_{5}) = (0, 0, 0, 0, 4) is a solution to x_{1} + x_{2} + x_{3} + x_{4} + x_{5} = 4.