There are 30,240 ways in which 5 men and 7 women can fill a row of 12 seats if the men sit on both ends.
How to solve permutation
If the men sit on both ends of the row, consider them as fixed positions. Therefore, arrange the remaining 3 men and 7 women in the remaining 10 seats.
To calculate the number of ways to arrange the men and women, use the concept of permutations.
The number of ways to arrange the 3 men in the 3 available seats is 3!.
Similarly, the number of ways to arrange the 7 women in the 7 available seats is 7!.
Since these arrangements are independent of each other, multiply the number of ways to arrange the men and women to get the total number of arrangements:
Total number of arrangements = 3! * 7!
= 6 * 5040
= 30,240
Therefore, there are 30,240 ways in which 5 men and 7 women can fill a row of 12 seats if the men sit on both ends.