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Chapter 11: Problem 70

Performing at a concert are eight rock bands and eight jazz groups. How many ways can the program be arranged if the first, third, and eighth performers are jazz groups?

Short Answer

Expert verified
The required number of ways is \( \binom{8}{3} \times 8! \times 8! \)

Step by step solution

01

Understanding the problem

We have 16 performance groups in total: 8 rock bands and 8 jazz groups. However, we've been told that the first, third, and eighth performers must be jazz groups.
02

Arranging the Jazz Groups

Calculate the permutations for the Jazz groups that have been assigned the 1st, 3rd, and 8th positions, and the remaining 5 jazz groups. We choose the groups for the three positions out of 8 jazz groups. This can be done in \( \binom{8}{3} \) ways. Then, we have to arrange these 8 jazz groups. Since they are distinct, they can be arranged in 8! ways.
03

Arranging the Rock Bands

Calculate the permutations for the 8 rock bands. Since they are distinct, they can be arranged in 8! ways.
04

Total arrangements

Since the jazz groups and rock bands arrangements are independent, we multiply the number of arrangements for the total number of ways in which the program can be arranged.

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