Shivani is planning a trip to Europe. She has a list of 10 countries she would like to see, but doesn't have time to visit all of them. She asks a travel agent to make an itinerary that includes any 3 of the 10 countries. How many sequences of 3 countries are possible?
Question
Shivani is planning a trip to Europe. She has a list of 10 countries she would like to see, but doesn't have time to visit all of them. She asks a travel agent to make an itinerary that includes any 3 of the 10 countries. How many sequences of 3 countries are possible?
Solution
This is a problem of permutations without repetition. In this case, the order in which Shivani visits the countries matters.
The formula for permutations without repetition is:
P(n, r) = n! / (n-r)!
where:
- n is the total number of options,
- r is the number of options chosen at a time,
- "!" denotes a factorial, meaning the product of all positive integers up to that number.
In this case, n = 10 (the total number of countries Shivani could visit) and r = 3 (the number of countries she will visit).
So, the calculation would be:
P(10, 3) = 10! / (10-3)!
= 10987654321 / 765432*1
= 1098
= 720
So, there are 720 different sequences of 3 countries that the travel agent could suggest to Shivani.
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