In how many ways can 10 engineers and 4 doctors be seated at a round table if all the 4 doctors do not sit together? Choose the correct option.
A) 13! - (10! × 4!) B) 13! × 4! C) 14! D) 10! × 4! E) None of these
Explanation :   Initially let's find out the total number of ways in which 10 engineers and 4 doctors   can be seated at a round table   In this case, n = total number of persons = 10 + 4= 
<gadget id="calculator">10 + 4</gadget>
<output>14</output>
14   Hence the total number of ways in which 10 engineers and 4 doctors can be seated at a round table= (14-1)= 
<gadget id="calculator">(14-1)</gadget>
<output>13</output>
13! ---(A)   Now let's find out the total number of ways in which 10 engineers and 4 doctors   can be seated at a round table where all the 4 doctors sit together.   Since all the 4 doctors sit together, group them together and consider as a single doctor.   Hence, n= total number of persons = 10 + 1= 
<gadget id="calculator">10 + 1</gadget>
<output>11</output>
11   These 11 persons can be seated at a round table in  (11-1)= 
<gadget id="calculator"> (11-1)</gadget>
<output>10</output>
10! ways ---(B)   However these 4 doctors can be arranged among themselves in 4! Ways ---(C)   From (B) and (C), the total number of ways in which 10 engineers and 4 doctors   can be seated at a round table where all the 4 doctors sit together= 10! 4! ---(D)   From (13! - (10! × 4!)) and (D),   The total number of ways in which 10 engineers and 4 doctors can be seated at a round table   if all the 4 doctors do not sit together = 13! - (10! 4!)   13! - (10! × 4!)nswer : 13! - (10! × 4!)
<result>A</result>
In how many ways can 10 engineers and 4 doctors be seated at a round table if all the 4 doctors do not sit together?