.
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
Solved.