How many ways can 5 students be seated in a row of 8 chairs? (2016)

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Q: How many ways can 5 students be seated in a row of 8 chairs? (2016)

Solution: The number of ways to seat 5 students in 8 chairs is 8P5 = 6720.

Steps: 9

Step 1: Understand that we have 8 chairs and we want to seat 5 students in those chairs.
Step 2: Realize that the order in which the students are seated matters. This means we are looking for permutations, not combinations.
Step 3: Use the permutation formula, which is written as nPr, where n is the total number of items (chairs) and r is the number of items to choose (students). In this case, n = 8 and r = 5.
Step 4: The formula for permutations is nPr = n! / (n - r)!. Here, '!' means factorial, which is the product of all positive integers up to that number.
Step 5: Calculate 8P5 using the formula: 8P5 = 8! / (8 - 5)! = 8! / 3!.
Step 6: Calculate 8! (which is 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1) and 3! (which is 3 × 2 × 1).
Step 7: Simplify the calculation: 8! = 40320 and 3! = 6, so 8P5 = 40320 / 6.
Step 8: Perform the division: 40320 / 6 = 6720.
Step 9: Conclude that there are 6720 different ways to seat 5 students in 8 chairs.