A password has to use the following Question

Question

A password has to use the following format: LDDDDLLL, where L can be any of the upper case letters in the alphabet, and D can be any digit including 0. None of the digits or letters can repeat.
What is the probability that a random password uses only vowels (A,E,I,O,U) and odd numbers (1,3,5,7,9)? Show work but do not evaluate.
What is the probability that the password spells the word “MATH?” Show work but do not evaluate.

expertsolution · Answer

please find the attachment. please complete full stars reviews as have done this with much efforts. read this and develop knowledge prior to use this. if you have any query just send me a message.  thanks
best of luck :)
regards,
dr. charles kennedy (expertsolution)

ivsivs · Answer

hey student,
note: please read and understand the answer file
the answer file consists of
complete answers and explanations to 2 problems in probability
the answers and explanations are complete
the answers and explanations have been properly solved, correct and checked.
thanks
question:
a password has to use the following format: lddddlll, where l can be any of the upper case letters in the alphabet, and d can be any digit including 0. none of the digits or letters can repeat. what is the probability that a random password uses only vowels (a,e,i,o,u) and odd numbers (1,3,5,7,9)?  show work but do not evaluate.
answer:
the password format is lddddlll, where the first and last characters must be letters, and the four characters in the middle must be digits.
since the password must contain only vowels and odd numbers, there are 5 vowels and 5 odd numbers to choose from, making a total of 10 possible characters. the first and last characters of the password can be any of the 5 vowels, so there are 5 options for each of those characters.
for the four digits in the middle, there are 5 odd numbers to choose from for each digit, and we can't repeat any digits. so the number of options for choosing the digits is 543*2 = 120.
finally, we need to choose 2 letters from the remaining 21 letters of the alphabet (after excluding the 5 vowels we used earlier). the order in which we choose the letters doesn't matter, so we can use the combination formula:
(21 choose 2) = 210
putting everything together, the total number of possible passwords is:
5 * 120 * 5 * 210 * 5 = 15,750,000
since there are no repeating characters in the password, each of these passwords is equally likely.
now we need to count how many of these passwords use only vowels and odd numbers. the first and last characters are vowels, so they are already taken care of. for the middle four digits, each digit can only be odd, so there are 5 options for each digit. therefore, the number of possible passwords that use only vowels and odd numbers is:
5 * 5 * 5 * 5 = 625
so the probability that a randomly chosen password uses only vowels and odd numbers is:
625 / 15,750,000 = 0.000039683
this is a very small probability, but that's to be expected given the strict requirements for the password.
question:
what is the probability that the password spells the word “math?”  show work but do not evaluate.
answer:
to spell the word "math" using the given format, we need to have m as the first letter, a as the second, t as the third, and h as the fourth. the probability of each letter appearing in the correct position depends on the number of choices available for each position.
for the first letter, we have 26 choices (the 26 letters in the alphabet), so the probability of getting m as the first letter is 1/26.
for the second letter, we have 25 choices (all the remaining letters in the alphabet after m has been used), so the probability of getting a as the second letter is 1/25.
for the third letter, we have 8 choices (the 10 digits 0-9, minus the two digits used in the first two positions), so the probability of getting t as the third letter is 1/8.
finally, for the fourth letter, we have 7 choices (the remaining 7 letters in the alphabet after m and a have been used), so the probability of getting h as the fourth letter is 1/7.
the total probability of getting the password in the desired format is the product of these probabilities:
1/26 * 1/25 * 1/8 * 1/7 = 1/327600
therefore, the probability of the password spelling the word "math" is 1 in 327,600.

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A password has to use the following Question

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