A tourist in France wants to visit 10 different cities. How many different routes are possible?
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To find the number of different routes a tourist can take to visit 10 different cities in France, you can use the concept of permutations.
A permutation is an arrangement of objects in a particular order. In this case, the cities will be the objects and the order in which the tourist visits them will be the arrangement.
To calculate the number of permutations, you can use the formula for permutation of n objects taken r at a time, which is denoted as P(n, r). In this case, the tourist wants to visit all 10 cities, so r = 10.
The formula for permutation is:
P(n, r) = n! / (n - r)!
Where n! denotes the factorial of n, which is the product of all positive integers less than or equal to n. Exclamation mark (!) represents factorial.
Applying this formula, we can calculate the number of different routes as follows:
P(10, 10) = 10! / (10 - 10)!
= 10! / 0!
= 10! / 1
= 10!
Simplifying further, we have:
10! = 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
= 3,628,800
Therefore, there are 3,628,800 different routes that the tourist can take to visit 10 different cities in France.