Let's calculate the number of ways the panel of judges can sit in a row for each scenario:
a. No restrictions:
In this case, we can simply calculate the total number of possible arrangements of the 6 judges in a row.
Number of ways = 6!
(6 factorial, which is 6 × 5 × 4 × 3 × 2 × 1)
Number of ways = 720
b. Men and women must alternate:
To satisfy this condition, we can fix the position of one gender (either men or women) and then arrange the other gender in the remaining positions.
Number of ways = 3! × 3!
(Number of ways to arrange the men) × (Number of ways to arrange the women)
Number of ways = 6 × 6
Number of ways = 36
c. Men must sit together and women must sit together:
In this scenario, we can treat the group of men and the group of women as single entities and arrange them in the row.
Number of ways = 2! × 3! × 3!
(Number of ways to arrange the two groups) × (Number of ways to arrange the men within their group) × (Number of ways to arrange the women within their group)
Number of ways = 2 × 6 × 6
Number of ways = 72
So, the number of ways the panel of judges can sit in a row is:
a. No restrictions: 720
b. Men and women must alternate: 36
c. Men must sit together and women must sit together: 72