2. Standard license plates in the state of New Jersey have three digits followed

Question

2. Standard license plates in the state of New Jersey have three digits followed by three letters (a) How many possible plates exist? (b) How many plates exist if neither digits nor letters can be repeated? (c) How many plates exist if the only restra tion is that letters and numbers cannot be adjacently placed? : i81 FRF is okay but 118-FFR is not) (d) What is the probability of choosing a random license plate whose letters spell out the word "MOM"? (e) What is the probability of choosing a random plate whose letters spell out cither "MOM" or "DAD? (O How many total possible plates would there be if the Department of Transportation of New Jersey decided to allow cither 3 digi or 3 letters followed by 3 digits?

Explanation / Answer

Note that there are 10 numbers ( 0 to 9) and 26 digits ( from A to Z)

2. Plate as 3 digits and 3 letters

a. The number of ways the plate can exist is 10^3 * 26^3

= 17576000

b. The number of ways the plate can exist with no rep. of letter or digit is

= 10*9*8*26*25*24

= 11232000

c. So same no. or letter can't be placed adj to each other

So, if u place a digit in 1st place, then you can place any one of the remaining 9 in 2nd place, then again in

10*9*9*26*25*25

= 13162500 ways

d. So, MOM is spelled out in XYZ-MOM or XYZ can have any of 10 no. in each of the 3 places.

So, 10^3 / (10^3 * 26^2)

= 1/26^3 is the answer

e. Like, DAD comes up with same probability. So, MOM or DAD come up with probability.

= 1/26^3 + 1/26^3

= 2/26^3

f. That would be 10^3 *26^3 times 2, as letter or digits can come first and then the other. So, 35152000 ways

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