A security code consists of either five or six different letters. How many distinct security codes are possible?
26P5 + 26P6
= 173 659 200
173659200
Well, I'm glad you asked. Let's do some code-cracking math, shall we?
If we have a security code consisting of 5 letters, we have 26 choices for the first letter, 25 choices for the second (since it can't be the same as the first), 24 choices for the third, 23 choices for the fourth, and 22 choices for the fifth.
So, the total number of distinct security codes with 5 letters would be 26 * 25 * 24 * 23 * 22 = 6,498,480.
Now, if we have a security code with 6 letters, the logic is the same except we have an extra choice. So, it would be 26 * 25 * 24 * 23 * 22 * 21 = 27,132,600.
Therefore, the total number of distinct security codes possible would be 6,498,480 + 27,132,600 = 33,631,080.
And just a reminder, this calculation assumes that each letter can only be used once in the security code. So, no repeating letters, please!
To find the number of distinct security codes possible, we need to consider two scenarios: one with five different letters and one with six different letters.
Scenario 1: Five different letters
In this case, we have 26 options for the first letter, 25 options for the second letter, 24 options for the third letter, 23 options for the fourth letter, and 22 options for the fifth letter. Thus, the number of distinct security codes with five different letters is calculated as:
26 * 25 * 24 * 23 * 22 = 789,360
Scenario 2: Six different letters
Similar to scenario 1, we have 26 options for the first letter, 25 options for the second letter, 24 options for the third letter, 23 options for the fourth letter, 22 options for the fifth letter, and 21 options for the sixth letter. Thus, the number of distinct security codes with six different letters is calculated as:
26 * 25 * 24 * 23 * 22 * 21 = 185,794,560
Therefore, the total number of distinct security codes possible is the sum of the possibilities from both scenarios:
789,360 + 185,794,560 = 186,583,920
So, there are 186,583,920 possible distinct security codes.