7 people sit at the round table. In how many ways can we sit them so that Bob and Bill don't sit opposing each other?  Choose the most appropriate option:  A) A)400  B) B)420  C) C)440  D) D)460  E) E)480
7 People: Bob,Bill,Red,Blue,White,Pink,Purple   Round Table: T   7 Chairs: 1,2,3,4,5,6,7   Let chairs 1 and 4 be opposite each other;   Bob is sitting on 1 and Bill is sitting on 4;   The rest of the people on 2,3,5,6,7 - can rearrange themselves in 5! ways   Now, Bill and Bob swap their positions;   Bob is sitting on 4 and Bill is sitting on 1;   The rest of the people on 2,3,5,6,7 - can rearrange themselves in 5! ways   Thus total number of ways in which Bob and Bill are sitting opposite each other = 2*5! = 120*2= 
<gadget id="calculator">120*2</gadget>
<output>240</output>
240   We have to find in how many ways they are NOT sitting opposite; subtract 240 from the total number of possible arrangements;   In circular permutation, Total number of arrangements of n people= (n-1)!   Here number of people = 7   Arrangements = (7-1)= 
<gadget id="calculator">(7-1)</gadget>
<output>6</output>
6!= 720   Number of ways Bob and Bill are NOT sitting opposite = 720-240= 
<gadget id="calculator">720-240</gadget>
<output>480</output>
480.   Answer: 480
<result>E</result>
7 people sit at the round table. In how many ways can we sit them so that Bob and Bill don't sit opposing each other?