At a music festival, there are ten bands scheduled to play, numbered 1 through 10. a. How many different ways can these bands be arranged to perform? b. If band 4 is performing first and band 7 last, then how many ways can their appearances be scheduled? a. There are $\square$ different ways to arrange the bands. (Simplify your answer.) b. If band 4 is performing first and band 7 last, there are $\square$ different ways to arrange the bands. (Simplify your answer.)

Step 1 ：There are 10 bands scheduled to perform at a music festival, numbered 1 through 10.

Step 2 ：To find the number of different ways these bands can be arranged to perform, we need to calculate the number of permutations of 10 bands.

Step 3 ：The number of permutations of 10 bands is \(10!\), which simplifies to \(\boxed{3628800}\) different ways.

Step 4 ：If band 4 is performing first and band 7 is performing last, we need to find the number of permutations of the remaining 8 bands.

Step 5 ：The number of permutations of 8 bands is \(8!\), which simplifies to \(\boxed{40320}\) different ways.