Probability - Gravner-9

Question

A group consists of $3$ Norwegians, $4$ Swedes, and $5$ Finns, and they sit at random around a table. What is the probability that all groups end up sitting together?

Answer 1

Number of Permutations among Norwegians = $3!$

Number of Permutations among Swedes = $4!$

Number of Permutations among Finns= $5!$

Number of permutations among these 3 groups = $(3-1)! = 2$

Favorable permutations = $3! \  4! \  5!  \ 2!$

Probability = $\large \frac{3! \  4! \  5!  \ 2!}{(12-1)!}=  \frac{3! \  4! \  5!  \ 2!}{11!}$