Question

34 bands compete to get showtime in a concert. The bands will either get the first showtime, the second showtime, or be the main show. How many different ways could the concert be set up using these 34 bands?

Answer 1

To determine the number of different ways the concert can be set up with 34 bands and 3 showtimes, calculate permutations for 34 items taken 3 at a time using the formula P(34, 3) = 34! / (34-3)!, resulting in 34 * 33 * 32 which equals 35,952 different setups.

Step-by-step explanation:

The question presented involves finding how many different ways a concert can be set up with 34 bands, given that there are three distinct showtimes available: first showtime, second showtime, and the main show. This is a combinatorial problem where we need to calculate the number of permutations of 34 bands taking 3 at a time, because the order in which the bands are chosen matters.

To solve this, we can use the formula for permutations which is P(n, r) = n! / (n-r)!, where n is the total number of items to choose from, r is the number of items to pick, and '!' represents factorial. In this case, n = 34 and r = 3.

The permutation calculation would be:

1. Calculate the factorial of 34, which is the product of all positive integers up to 34.
2. Calculate the factorial of 31, which is the product of all positive integers up to 31.
3. Divide the factorial of 34 by the factorial of 31 to find the number of permutations, P(34, 3).

Performing the calculation, 34! / (34-3)! simplifies to 34 * 33 * 32, since the terms from 1 to 31 in the factorial will cancel out.

The total number of different ways the concert can be set up is 34 multiplied by 33 multiplied by 32, which equals 35,952 setups.