A group of ten students includes triplets Alex, Alison and Alice wearing black, white and yellow t-shirts respectively. They are to be seated in groups of five around two tables: one green and one red. The triplets insist on sitting together. Determine the number of seating arrangements possible.
Choosing the groups The group with the triplets need two additional people to be chosen in: $${10-3 \choose 2} ={7 \choose 2}=21 \text{ Ways}$$
Seating the non-triplet group Five people can sit around a circular table in $$(5-1)!=4!=24 \text{ Ways}$$
Seating the triplets Since the triplets must sit together, consider them as one person: $$5-2=3 \text{ people}\implies (3-1)!=2!=2 \text { circular arrangements}$$ The triplets are distinguishable and can be arranged among themselves in $3!=6 \text{ ways}$. The total number of ways of seating the triplets, by the multiplication principle is: $$2\times 6=12$$
Choosing the table We have the following two choices: {Green: Triplet, Red: Non-triplet} {Red: Triplet, Green: Non-triplet} $$2 \text{ ways}$$
Final Answer The final answer is then: $$21\times 24 \times 12 \times2 =12,096$$
Is this correct?