IR-TST 2017 Problem 11

$k,n$ are two arbitrary positive integers. Prove that there exists at least $(k-1)(n-k+1)$ positive integers that can be produced by $n$ number of $k$ 's and using only $+,-,\times, \div$ operations and adding parentheses between them, but cannot be produced using $n-1$ number of $k$ 's.

Proposed by Aryan Tajmir