If 6 people are going to sitting at a round table, but Sam will not si

Hi pclawong

i think i can help you with this.

first of all if 6 people are sitting around a table, and the condition is Sam will not sit next to Suzie.

this will be equal to total no. of ways 6 people can sit around a table - total no of ways in which Sam will always sit next to Suzie

now total no of ways, 6 people can sit around a circular table is (6-1)!
and total no of ways, Sam will always sit next to Suzie is (5-1)! 2!

Note: total no. of ways in which n different things can be arranged around a circular path is (n-1)!

now let me tell you why we are doing (5-1)! 2! for the later case-

in later case we are assuming that Sam will always sit next to Suzie, so we are considering them as a single unit saying that now we have 5 people not 6.

now number of way, 5 people can sit around a circular table is (5-1)!

now again Sam & Suzie can interchange their position so we are doing 2! for them

so now total no. of ways, Sam will always sit next to Suzie is (5-1)! 2!

now the question is asking about ways in which Sam will not sit next to Suzie.

we subtract (5-1)! 2! into total no of ways which is (6-1)!

(6-1)! - (5-1)! 2!

hope this will help you.

Thank you GTExl
I get it now after combined you both explanation.
God bless you

6 people can be arranged in (6-1)!=120
let sam and Suzie be sitting together = (5-1)!*2=4!*2=48
Ways they don't sit together=120-48=72

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