Question 1184938
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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?
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<pre>
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.        <U>ANSWER</U>
</pre>

Solved.