Matt must take a new code for the lock to his deer lease. It must consist of 3 letters, followed by 5 digits, then 2 letter and 1 more digit. The first letter must be a vowel and the middle digit of the first 5 must be even. If no letter or digit can be repeated, how may code words can be formed?
5x26x25 x10x9x5x8x7 x24x23x6= 2.713x10^11 Is this correct ?
since there are no repeats,
3 letters, 1st vowel: 5*25*24
5 digits, middle even: 5*9*8*7*6
(consider picking the even digit first)
2 letters: 23*22
1 digit: 5
so I get only 114,768,800,00
To determine the number of possible code words that can be formed, we need to break down the problem into separate parts.
1. First, let's consider the possibilities for the three letters at the beginning of the code. The first letter must be a vowel, so there are 5 vowels (A, E, I, O, U) to choose from. Since no letter can be repeated, we have 5 choices for the first letter. For the second and third letters, we have 25 choices each (26 letters minus the vowel already chosen). Therefore, the total number of possibilities for the first three letters is 5 x 25 x 25.
2. Next, let's consider the five digits in the middle of the code. The middle digit must be even, so we have 5 choices (0, 2, 4, 6, 8) for that digit. For the remaining four digits, we have 10 choices for each (0-9), but we cannot repeat any digit. So, for the second digit, we have 9 choices (since we used one digit already), for the third digit, we have 8 choices (since we used two digits already), and so on. Therefore, the total number of possibilities for the five digits is 5 x 10 x 9 x 8 x 7.
3. Lastly, let's consider the last two letters and the final digit in the code. Since no letter or digit can be repeated, we have 24 choices for the first of the last two letters and 23 choices for the second. For the final digit, we have 6 choices (0, 2, 4, 6, 8) since it can be any even number except the middle digit. Therefore, the total number of possibilities for the last two letters and the final digit is 24 x 23 x 6.
To calculate the total number of possible code words, we multiply the individual possibilities together:
Total possibilities = (5 x 25 x 25) x (5 x 10 x 9 x 8 x 7) x (24 x 23 x 6)
Calculating this expression gives us a total of 1,380,000 possibilities. Therefore, the answer is not correct. It should be 1,380,000, not 2.713x10^11.