Over the past week, an airline pilot was n Vancouver three days, Calgary two days, Whitehorse one day, and Winnipeg two days.
a) How many different itineraries could the pilot have had?
b) If you knew that the two days in Calgary were consecutive (a layover), how many itineraries could the pilot have had?
I figured out A..
8! / 3!2!1!2! = 1680 different itineraries..
but i don't understand what the consecutive part for B means.
In effect what you were doing in a) was to find the number of arrangements of
V,V,V,C,C,WH,WI,WI
What it says in b) is that the C,C have to be side by side in the arrangement.
One way to do this is to consider C,C as a single element, so now you have 7 things to arrange instead of 8, you still have 3 V's, 2 WI's
so for b)
no of ways = 7!/(3!2!) = 420
In the given scenario, the pilot has visited Vancouver, Calgary, Whitehorse, and Winnipeg for a certain number of days. In order to determine the number of different itineraries, we can use permutations and combinations.
a) To find the number of different itineraries without any restrictions, we can consider the total number of days spent in each location. The formula for finding the number of permutations with repetition is n!/n1!n2!n3!...nr!, where n is the total number of elements and n1, n2, n3,..., nr are the repetitions of each element.
In this case, we have 8 days in total, with 3 days in Vancouver, 2 days in Calgary, 1 day in Whitehorse, and 2 days in Winnipeg. Using the formula mentioned above, the number of different itineraries would be:
8! / (3! * 2! * 1! * 2!) = 1680 different itineraries.
b) Now, let's consider the scenario where the pilot had two consecutive days in Calgary. This means that the pilot spent two consecutive days in Calgary without visiting any other destination in between.
To approach this, we can treat the consecutive two days in Calgary as a single entity, which means we will now have only 7 elements to arrange. In this case, the formula for finding the number of permutations without repetition will be used, and the number of itineraries would be:
7! / (3! * 1! * 1! * 2!) = 420 different itineraries.
The consecutive part in this case indicates that the pilot had two days in Calgary together, without any other destinations in between those two days.