An ATM personal identification number (PIN) consistsof four digits, each a 0, 1, 2,… 8, or 9, in succession.a. How many different possible PINs are there if thereare no restrictions on the choice of digits?b. According to a representative at the author’s localbranch of Chase Bank, there are in fact restrictions onthe choice of digits. The following choices are prohibited:(i) all four digits identical (ii) sequences ofconsecutive ascending or descending digits, such as6543 (iii) any sequence start ing with 19 (birth yearsare too easy to guess). So if one of the PINs in (a) israndomly selected, what is the probability that it willbe a legitimate PIN (that is, not be one of the prohibitedsequences)?c. Someone has stolen an ATM card and knows thatthe first and last digits of the PIN are 8 and 1,respectively. He has three tries before the card isretained by the ATM (but does not realize that). Sohe randomly selects the 2nd and 3rd digits for thefirst try, then randomly selects a different pair ofdigits for the second try, and yet another randomlyselected pair of digits for the third try (the individualknows about the restrictions described in (b) soselects only from the legitimate possibilities). Whatis the probability that the individual gains access tothe account?d. Recalculate the probability in (c) if the first and lastdigits are 1 and 1, respectively.
Permutations and Combinations
If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.
Counting Theory
The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.
An ATM personal identification number (PIN) consists
of four digits, each a 0, 1, 2,… 8, or 9, in succession.
a. How many different possible PINs are there if there
are no restrictions on the choice of digits?
b. According to a representative at the author’s local
branch of Chase Bank, there are in fact restrictions on
the choice of digits. The following choices are prohibited:
(i) all four digits identical (ii) sequences of
consecutive ascending or descending digits, such as
6543 (iii) any sequence start ing with 19 (birth years
are too easy to guess). So if one of the PINs in (a) is
randomly selected, what is the
be a legitimate PIN (that is, not be one of the prohibited
sequences)?
c. Someone has stolen an ATM card and knows that
the first and last digits of the PIN are 8 and 1,
respectively. He has three tries before the card is
retained by the ATM (but does not realize that). So
he randomly selects the 2nd and 3rd digits for the
first try, then randomly selects a different pair of
digits for the second try, and yet another randomly
selected pair of digits for the third try (the individual
knows about the restrictions described in (b) so
selects only from the legitimate possibilities). What
is the probability that the individual gains access to
the account?
d. Recalculate the probability in (c) if the first and last
digits are 1 and 1, respectively.
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