66. Access Code An access code consists of six characters. For each character, any letter or number can be used, with the exceptions that the first character cannot be 0 and the last two characters must be odd numbers.
a. What is the probability of randomly selecting the correct access code on the first try?

Verified step by step guidance

1. Step 1: Determine the total number of possible characters for each position in the access code. There are 36 possible characters (26 letters + 10 digits) for most positions, except for the first and last two positions which have specific restrictions.
2. Step 2: For the first character, it cannot be '0', so there are 35 possible choices (26 letters + 9 digits excluding '0').
3. Step 3: For the last two characters, they must be odd numbers. There are 5 odd digits (1, 3, 5, 7, 9), so each of these two positions has 5 possible choices.
4. Step 4: For the remaining three middle characters, there are no restrictions, so each of these positions has 36 possible choices.
5. Step 5: Calculate the total number of possible access codes by multiplying the number of choices for each position: (35 choices for the first character) × (36 choices for the second character) × (36 choices for the third character) × (36 choices for the fourth character) × (5 choices for the fifth character) × (5 choices for the sixth character). Then, calculate the probability of randomly selecting the correct access code on the first try by taking the reciprocal of this total number of possible access codes.

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics dealing with counting, arrangement, and combination of objects. In this context, it helps determine the total number of possible access codes by considering the restrictions on character selection, such as the first character not being '0' and the last two characters being odd numbers.

Probability

Probability is a measure of the likelihood that an event will occur, expressed as a ratio of favorable outcomes to the total number of possible outcomes. To find the probability of selecting the correct access code on the first try, one must calculate the ratio of successful outcomes (1 correct code) to the total number of valid access codes.

Permutations and Restrictions

Permutations refer to the different ways in which a set of items can be arranged. In this scenario, the restrictions on the first character and the last two characters create specific conditions that must be accounted for when calculating the total number of valid access codes, impacting the overall probability.