Solution:

Step 1: Find the total number of permutations of 10 different things taken 4 at a time.

The number of permutations of n different things taken r at a time is given by nPr = n!/(n-r)!. Therefore, the number of permutations of 10 different things taken 4 at a time is:

10P4 = 10!/(10-4)! = 10!/6! = 10 x 9 x 8 x 7 = 5,040

Step 2: Find the number of permutations of 9 different things taken 4 at a time.

If one particular thing never occurs, then we have only 9 things to choose from. Therefore, the number of permutations of 9 different things taken 4 at a time is:

9P4 = 9!/(9-4)! = 9!/5! = 9 x 8 x 7 x 6 = 3,024

Step 3: Subtract the number of permutations of 9 different things taken 4 at a time from the total number of permutations of 10 different things taken 4 at a time.

The number of permutations of 10 different things taken 4 at a time in which one particular thing never occurs is:

10P4 - 9P4 = 5,040 - 3,024 = 2,016

Therefore, there are 2,016 permutations of 10 different things taken 4 at a time in which one particular thing never occurs.

Conclusion:

The number of permutations of 10 different things taken 4 at a time in which one particular thing never occurs is 2,016.