Answer to Question #101592 in Combinatorics | Number Theory for Nahsor

Question

How many 5 letter code words (not necessarily word with sense) can be formed from the letter A B C D E F G H I and J if ? a. The word has at least one vowel

Answer 1

We may count the solution if we take the whole amount of 5 letter code words out of 10 unique letters (that may be used only once) subtracted by the amount of 5 letter code words out of letters that have no vowels at all(there are 7 of them).

Thus, the whole amount of possible words is 10! / (10-5)!

To get that, we may check this:

We have 5 letters to have in a code. For the first letter, we have 10 options (letters). For the second one, we have 9 (because we chose one letter for the first letter of the code). Going further, the whole amount is:

10 * 9 * 8 * 7 * 6 = 10! / (10-5)!

To count the amount of words that have no vowels in them, we take 7 letters ( B C D F G H J ) as the available ones and the result is:

7 * 6 * 5 * 4 * 3 = 7! / (7-5)!

So, the answer is 10*9*8*7*6 - 7*6*5*4*3 = 7*6*5*4*3(2*3*2 - 1) = 2520*11 = 27720.