A newly designed international airport is planning to label its gates. Each gate must be assigned a letter from the alphabet from A to M (13 in all), and a number from 1 to 15. To avoid confusion, any two gates must be assigned a different letter, or a different number (or both). What is the maximum number of distinct gates that the airport can have?
To find the maximum number of distinct gates that the airport can have, we need to consider the total number of possible combinations for the letters and numbers.
First, let's calculate the number of possibilities for the letters. We have 13 letters available from A to M. Therefore, there are 13 possibilities for the first letter, 12 for the second letter, 11 for the third, and so on, until we have assigned all the letters. This can be calculated using factorial notation: 13! (13 factorial).
Next, let's calculate the number of possibilities for the numbers. We have 15 numbers available from 1 to 15. Therefore, there are 15 possibilities for the first number, 14 for the second number, and so on, until we have assigned all the numbers. This can also be calculated using factorial notation: 15! (15 factorial).
To find the maximum number of distinct gates, we need to multiply the total number of possibilities for the letters by the total number of possibilities for the numbers. Thus, the maximum number of distinct gates can be calculated as:
13! × 15!
Now, let's calculate the value:
13! × 15! ≈ 6.74 x 10^26
Therefore, the maximum number of distinct gates that the airport can have is approximately 6.74 x 10^26.