3.12 Combinatorics

E129

[09N] Let be given \(n,k\) natural with \(k≥ 1\). How many different choices of vectors \((j_ 1,\ldots j_ k)\) of natural numbers are there such that \(j_ 1+\cdots +j_ k=n\) ? How many different choices of vectors \((j_ 1,\ldots j_ k)\) of positive natural numbers are there such that \(j_ 1+\cdots +j_ k=n\) ? Hidden solution: [UNACCESSIBLE UUID ’09P’]

E129

[09Q] Let \(n,m\) be positive integers and let \(I=\{ 1,\ldots ,n\} ,J=\{ 1,\ldots m\} \).

• How many functions \(f:I→ J\) are there?

• How many functions \(f:I→ J\) are injective?

• How many functions \(f:I→ J\) are strictly growing?

• How many functions \(f:I→ J\) are weakly increasing?

Hidden solution: [UNACCESSIBLE UUID ’09R’]

See also exercise 3.