Count Number Of Time A Text String Appears

I have a very large spreadsheet and want to count the number of times a particular text string shows up in a column. I can't use autofilter due to the 1000 limit.

Here's an example, Column C contains:
Dan Parker
John Doe
Dan Smith
Jill Smith

So if I search on *Dan*, the function should return a count of 2.

I've used COUNTIF before to return values when the whole cell = a certain value but in my case the cell may have 200 characters and I want to count based on a fuzzy search. I would like to do this in a function and not a macro.