Five country music bands and 3 rock bands are signed up to perform at an all-day festival. How many different orders can the bands play in if the following conditions apply? a. there are no restrictions on the order
b. all the bands of each type must perform in a row

a. There are 6 country music bands and 3 rock music bands. Since, there are no restrictions on the order of play, we use combination theorem. So Combination, ⁿCₓ =  . The total number of bands n = 6 + 3 = 9. x = 1 , because we combine it in one way, since only one band can perform at a time.  ⁹C₁ = =9!/8!= 9 orders of play.

b. Since all the bands of each type must perform in a row, we use permutation theory. ⁿPₓ where n = number of bands and x = number of ways to be permuted. For the country music bands, n = 6 and x =1 (since only one band can perform at a time). ⁶P₁ = 6 orders of play. For the rock bands, n = 3 and x =1 (since only one band can perform at a time). So, ³P₁ = 3 orders of play. The total different number of orders is ⁶P₁ + ³P₁ = 6 + 3 = 9 orders of play.

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answer: a clock

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