A radio DJ is contractually obligated to run a commercial for Pepsi seven times, a commercial for a local law firm once, and a new promo for the station twice per hour. In how many different ways can he do this?
The DJ can run the commercials in different ways.
Answer
Final Answer: The DJ can run the commercials in \(\boxed{360}\) different ways.
Step 1 ：This is a permutation problem with repetition. The DJ has to run 10 commercials in total (7 for Pepsi, 1 for the law firm, and 2 for the station promo). However, since the commercials for Pepsi and the station promo are identical, we have to divide by the factorial of the number of these identical items to avoid over-counting.
Step 2 ：The total number of commercials is 10.
Step 3 ：The number of permutations is calculated as \(\frac{10!}{7! \times 1! \times 2!} = 360\).
Step 4 ：Final Answer: The DJ can run the commercials in \(\boxed{360}\) different ways.
