A class has 2 rows of 8 seats each. there are 14 students 4 of whom always sit in the front row and 5 of whom always sit in the back row. How many ways can the students be arranged?
To determine the number of ways the students can be arranged, we can break down the problem into four steps:
Step 1: Assign the students who always sit in the front row.
Since there are 4 students who always sit in the front row, we can consider it as a single group. We have to find the number of ways to arrange this group of 4 students. This can be calculated using the formula for permutations of a set, which is n!, where n is the number of elements in the set. In this case, n = 4, so there are 4! = 4 × 3 × 2 × 1 = 24 ways to arrange these 4 students in the front row.
Step 2: Assign the students who always sit in the back row.
Similar to Step 1, we now have to find the number of ways to arrange the 5 students who always sit in the back row. Again, using the permutation formula, we have 5! = 5 × 4 × 3 × 2 × 1 = 120 ways to arrange these 5 students in the back row.
Step 3: Assign the remaining students to the empty seats in the front row.
Now, we have 14 total students, out of which we have already assigned 4 students to the front row. Therefore, there are 14 - 4 = 10 remaining students to be placed in the front row. Since there are 8 seats in the front row, we need to find the number of ways to choose 10 students out of the remaining 10 to fill these seats. This can be calculated using the combination formula, which is written as nCr, where n is the number of elements in the set and r is the number of elements to be selected from the set. In this case, n = 10 and r = 8, so there are 10C8 = 10! / (8! * (10-8)!) = 45 ways to choose 10 students to fill 8 seats in the front row.
Step 4: Assign the remaining students to the empty seats in the back row.
Similar to Step 3, there are now 14 - (4+10) = 14 - 14 = 0 remaining students to be placed in the back row. Since there are no more students to assign to the back row, there is only 1 way to place these 0 students.
To find the total number of ways to arrange the students, we multiply the results from each step together:
Total number of ways = Number of ways to assign students in the front row × Number of ways to assign students in the back row × Number of ways to assign remaining students to the empty seats in the front row × Number of ways to assign remaining students to the empty seats in the back row.
Total number of ways = 24 × 120 × 45 × 1 = 129,600 ways.
Therefore, there are 129,600 different ways to arrange the students in the class.