1) License plates in the great state of Utah consist of 2 letters and 4 digits.
ID: 3829620 • Letter: 1
Question
1) License plates in the great state of Utah consist of 2 letters and 4 digits. Both digits and letters can repeat and the order in which the digits and letters matter. Thus, AA1111 and A1A111 are different plates. How many possible plates are there?
A)None of the above
B)26^2x10^4x15
C)36^6
2) Like in Utah, license plates in the great state of Wisconsin consist of 2 letter and 4 digits. However, only digits can repeat and the two letters always precede the digits. How many possible plates are there in Wisconsin?
A)26x26x10^4
B)26x25x10^4
C)36x35x10x10x10x10
3)An individual's chances of being killed by a lighting strike in a given year is 1/5,000,000. The Georgia Lottery has just invented a new game "Thunder" in which you win if you guess 5 out of 65 numbers right (order does not matter).
Are your chances of winning Thunder higher or lower than your chances of dying in a lightning strike in a given year?
A)Lower
B)The same
C)Cant compare
4) The cost of health insurance is $1,200 per year. The cost of an operation for a dramatic injury is $500,000. The chances of someone needing such an operation are 47.3% over a twenty year period. From a purely financial point of view, are you better off taking out the insurance or not?
A)No. The insurance company must make a profit
B)Yes. The cost of the insurance is less than the probability cost of the operation
C)No. The probability of needing the operation is less than 50%
5) Choose the truth table and associated logic that determines whether the following argment is valid:
If there is a tractor trailor accident on I-75, I-75 is blocked and highway 41 is blocked.
Highway 41 is not blocked.
Therefore, there is no tractor trailer accident on I-75.
A) The argument is not valid, as the following truth table shows, where
p is "There is a tractor trailer accident on I-75"
q is "Highway 41 is blocked"
and
r is "I-75 is blocked"
p | q | r | q & r | p (q & r) | ~q | ~p
1 | 1 | 1 | 1 | 1 | 0 | 0
1 | 1 | 0 | 0 | 0 | 1 | 0
1 | 0 | 1 | 0 | 0 | 0 | 0
1 | 0 | 0 | 0 | 0 | 1 | 0
0 | 1 | 1 | 1 | 1 | 0 | 1
0 | 1 | 0 | 0 | 1 | 1 | 1
0 | 0 | 1 | 0 | 1 | 0 | 1
0 | 0 | 0 | 0 | 1 | 1 | 1
An inspection of the table above shows that every line on which p (q & r) and ~q are true, ~p is not alwaystrue as well.
B) The argument is valid, as the following truth table shows, where
p is "There is a tractor trailer accident on I-75"
q is "Highway 41 is blocked"
and
r is "I-75 is blocked"
p | q | r | q & r | p (q & r) | ~q | ~p
1 | 1 | 1 | 1 | 1 | 0 | 0
1 | 1 | 0 | 0 | 0 | 0 | 0
1 | 0 | 1 | 0 | 0 | 1 | 0
1 | 0 | 0 | 0 | 0 | 1 | 0
0 | 1 | 1 | 1 | 1 | 0 | 1
0 | 1 | 0 | 0 | 1 | 0 | 1
0 | 0 | 1 | 0 | 1 | 1 | 1
0 | 0 | 0 | 0 | 1 | 1 | 1
An inspection of the table above shows that every line on which p (q & r) and ~q are true, ~p is true as well.
C) None of the above
Explanation / Answer
1)
C) 36^6
There is no condition given whether letters precede numbers or not. 111A1A and 1111AA are valid cases. So, we have 36 choices for each of the 6 places. 26 choices for A-Z and 10 choices for (0-9), so 26+10=36 choices. And we have 6 places overall and also the letters and digits can repeat. Hence, by fundamental principle of counting total number of plates are 36*36*36*36*36*36 ,i.e., 36^6.
2)
B)26x25x10^4
In this case alphabets cannot repeat and always precede the digits. And digits can repeat. So, the first two places in the plate are meant for letters. So, we have 26 choice for the first letter and the letter in the second place cannot be the same as the first one. So 25 choices for the second. And, 10 choices for each of the next 4 places. So total number of plates 26*25*10*10*10*10 ,i.e., 26*25*10^4.
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