fuctions with multiple conditions

i'm sure this is a repeated question, but i was wondering if someone could coach me through counting values in a column based on whether certain criteria were met in two (or more) other columns. basically, i have a table where each row represents a record. i want to count all the records that satisfy conditions in two of the fields.

i've been trying to do an array function like this:

count(if(and(a1:a10=1,b1:b10>date(2000,12,31)),a1:a10)

this will not give me an error, but i won't get the right answer either. any advice? thanks.