SOLUTION: A security code consists of four different letters chosen from the 26 in alphabet, order being important. For example, QCDK. Given that the code does not contain any vowels what i

Algebra ->  Permutations -> SOLUTION: A security code consists of four different letters chosen from the 26 in alphabet, order being important. For example, QCDK. Given that the code does not contain any vowels what i      Log On


   

Question 588172: A security code consists of four different letters chosen from the 26 in alphabet, order being important. For example, QCDK. Given that the code does not contain any vowels what is the probability that it will contain an X?
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
ONE WAY:
I would calculate the probability of no X and subtract from 1.
Not counting Y as a vowel, there are 5 vowels in the 26 alphabet letters, leaving 21 consonants to choose from.
The probability of not having an X as the first letter would be 20%2F21.
The probability of not having an X as the second letter would be 19%2F20, because whatever letter was chosen first would not be available.
The probability of not having an X as the third letter would be 18%2F19.
The probability of not having an X as the fourth letter would be 17%2F18.
The probability of not having an X would be

The probability of having one X would be
1-17%2F21=4%2F21
ANOTHER WAY:
There are 20%2A19%2A18 ways to choose a 3-letter combination with no repeated letters from the set of consonants that are not X.
Inserting an X in any of the 4 possible positions, you get the
4%2A20%2A19%2A18 combinations of four different consonants including one X.
There is a total of 21%2A20%2A19%2A18 combinations of four different consonants.
Among those, we had found that only 4%2A20%2A19%2A18 could include an X.
The fraction including an X is
4%2A20%2A19%2A18%2F%2821%2A20%2A19%2A18%29=4%2F21