TH-TMO 2014 Problem 8

Let $n$ be a positive integer. We want to make up a collection of cards with the following properties:

1. each card has a number of the form $m!$ written on it, where $m$ is a positive integer.

2. for any positive integer $t \le n!$ , we can select some card(s) from this collection such that the sum of the number(s) on the selected card(s) is $t$ .

Determine the smallest possible number of cards needed in this collection.