Multiple choice

A school has 11 Maths teachers and 7 Science teachers. In how many ways can they be seated in a round table so that all the Science teachers do not sit together?

  1. ${ 18! } -( { 12! } \times{ 7! } )  $
  2. ${ 17! } -( { 12! } \times{ 7! } )  $
  3. ${ 17! } -( { 11! } \times{ 7! } )  $
  4. ${ 18! } -( { 11! } \times{ 7! } )  $
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Total people = 18. Total arrangements in a circle = (18-1)! = 17!. Arrangements where all 7 Science teachers sit together: Treat them as 1 unit. Total units = 11 + 1 = 12. Arrangements = (12-1)! * 7! = 11! * 7!. Total ways = 17! - (11! * 7!).

AI explanation

First, find the total unrestricted arrangements of all 18 teachers around a round table, which is (18 - 1)! = 17!. Next, use the method of subtraction to remove the arrangements where all 7 Science teachers sit together. Treating the 7 Science teachers as a single block along with the 11 Maths teachers, we have 12 entities to arrange in a circle, yielding (12 - 1)! = 11! ways. The 7 Science teachers can be arranged within their block in 7! ways, so the non-together arrangements are calculated as 17! - (11! * 7!).

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