5 Couples, having dinner before the prom are to be seated at a circular table. In how many ways can this occur if the seating must be boy girl, boy girl?
To solve this problem, we will consider a few steps:
Step 1: Determine the number of ways to arrange the boys and girls alternately.
Since we have 5 couples, there are 5 boys and 5 girls. The first couple can be seated arbitrarily, so we can fix its position without loss of generality. Therefore, there are 4! (4 factorial) ways to arrange the remaining 4 boys and 4 girls alternately around the table.
Step 2: Determine the number of ways to arrange each couple within themselves.
For each couple, we can arrange the boy and girl in 2 ways (boy-girl or girl-boy).
Step 3: Calculate the total number of possibilities.
Since each of the 5 couples can be arranged independently, we multiply the number of possibilities for each couple together.
Total number of possibilities = (4!) * (2^5)
Step 4: Calculate the final answer.
Evaluate the expression (4!) * (2^5) to find the total number of ways the seating can occur.
4! = 4 x 3 x 2 x 1 = 24
2^5 = 2 x 2 x 2 x 2 x 2 = 32
Final answer = 24 * 32 = 768
So, there are 768 ways to seat the 5 couples at the circular table with boy-girl, boy-girl arrangements.