Permutation

We know that $4$ numbers are taken at a time. Hence there are $4$ places. The first place is always $1$ , therefore the first place can be filled up in $1$ way The next place can be filled with any of the numbers except $1$ , therefore, the second place can be filled up in $6$ ways. The next place can be filled up in $5$ ways because repetition is not allowed. Similarly, the last place can be filled up in $4$ ways.

Now by the Fundamental Principle of Counting (By Multiplication), the number of possible numbers are $1 \times 6 \times 5 \times 4 = \boxed{120}$