SOLUTION: In how many ways can 3 engineers,4 teachers,4 doctors,3 businessmen and 3 nurses can be seated in a row for COVID 19 check-up so that those of the same carrier sit together?

We have 5 groups according their professional occupation, and we can permute these 5 groups as whole objects
in 5! = 5*4*3*2*1 = 120 ways.


Next, we can permute by 3! = 6 ways inside the group of 3 engineers;

             permute by 4! = 24 ways inside the group of 4 teachers;

             permute by 4! = 24 ways inside the group of 4 doctors;

             permute by 3! =  6 ways inside the group of 3 businessmen;

             permute by 3! =  6 ways inside the group of 4 nurses.


All these permutations are independent, so the total number of permutations under the imposed condition is


    120*6*24*24*6*6 = 14929920.        ANSWER