How many 11 -digit positive integers can be written using only the digits 1,2 and 3 and have exactly five 1’s and exactly two 2's as digits?
We know there must be five 1's, two 2's, and four 3's.
There are \({11! \over 2! \times 5! \times 4!} = \color{brown}\boxed{6930}\) numbers that work.