Bunuel wrote:
A license plate number is created by writing a sequence of three letters followed by a sequence of 3 digits. Any digit 0 - 9 is acceptable, as is any letter. What is the probability that either (but not both) the letters or numbers form a palindrome, a series of letters or numbers that read the same forwards and backwards? (For example, ADA and 121 are palindromes but ABC and 123 are not. ADA121 would not be valid but ADA123 would be valid.)
A. 1/26
B. 1/10
C. 17/130
D. 7/52
E. 9/65
CASES Total
\(26*26*26*10*10*10 = {26}^3{10}^3\)
\(T.Letters = {26}^3\)
\(T.Digits = {10}^3\)
CASES Palindrome
ABA: 26*25*1 = 650
AAA: 26*1*1 = 26
010: 10*9*1 = 90
000: 10*1*1 = 10
\(P.Letters = 676 = {26}^2\)
\(P.Digits = 100 = {10}^2\)
CASES Either Palindrome Not Both
\([Palindrome.Letters]*[T.digits-100]: [{26}^2]*[{10}^3-{10}^2]\)
\([T.letters-650]*[Palindrome.Digits]: [{26}^3-{26}^2]*[{10}^2]\)
PROBABILITY: Favorable Cases / Total Cases
\(Favorable:[{26}^2]*[{10}^3-{10}^2]+[{26}^3-{26}^2]*[{10}^2]…{26}^2{10}^2(9+25)\)
\(Probability:{26}^2{10}^2(9+25)/{26}^3{10}^3…34/260…17/130\)
Ans (C)