The Futurists, a rock group, are planning a concert tour with performances to begiven in five cities: San Francisco, Los Angeles, San Diego, Denver, and Las Vegas.
How many ways can they arrange their itinerary if
a.There are no restrictions?
b.The tree performances in California must be given consecutively?
a. If there are no restrictions, they can go to any of the five cities first, then any of the remaining four, then any of the remaining three, etc.
Therefore, the answer would be \(5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120\) ways.
b. If they must give the performances in San Francisco, Los Angeles, and San Diego must be given consecutively, we can group the three together. They can either perform in one of the two other cities or in California, and then they can perform in the two other options they haven't done yet, and then the final option. We can order this in \(3 \cdot 2 \cdot 1 = 6\) ways.
However, we can also order the performances in California. They can perform in any of the three cities, then any of the remaining two, then the final option. From these, we also have \(3 \cdot 2 \cdot 1 = 6\) ways.
For every way we perform the three main groups, we can rearrange the performances in California. Therefore, we get a total of \(6 \cdot 6 = 36\) ways.