The digits 0, 1, 2, 3, 4, 5, 6 and 7 are used to make 4 digit codes. (a) How many unique codes are possible if the digits can be repeated? (b) How many unique codes are possible if the digits canot be repeated? (c) In the case where digits may be repeated, how many codes are numbers that are greater than 2 000 and even? In the case where digits may not be repeated, how many codes are () numbers that are greater than 2 000 and divisible by 4? (e) What is the probability that a code will contain at least one 7? The reneated
The digits 0, 1, 2, 3, 4, 5, 6 and 7 are used to make 4 digit codes. (a) How many unique codes are possible if the digits can be repeated? (b) How many unique codes are possible if the digits canot be repeated? (c) In the case where digits may be repeated, how many codes are numbers that are greater than 2 000 and even? In the case where digits may not be repeated, how many codes are () numbers that are greater than 2 000 and divisible by 4? (e) What is the probability that a code will contain at least one 7? The reneated
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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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.
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Transcribed Image Text:**Understanding Number Plate and Digit Code Combinations**
The image contains two sections of questions regarding number plates and digit codes. Below, each question is presented along with an explanation of the context where needed.
### Number Plate Questions
The upper section deals with a hypothetical change in the number plate system.
#### Question Set:
(a) **How many possible number plates were there with the old system?**
(b) **How many number plates are there with the new system?**
(c) **Why do you think that the Traffic Department changed to the new system?**
**Explanation:** The diagram associated with these questions shows a number plate example labeled "BB 01 BB GP" suggesting a format where letters and digits are used together in a certain pattern. The reasons for changing the system might be due to an intention to increase the number of possible number plates to accommodate more vehicles.
### Digit Code Questions
The lower section focuses on generating 4 digit codes using specific digits.
The digits allowed are: **0, 1, 2, 3, 4, 5, 6, and 7**
#### Question Set:
(a) **How many unique codes are possible if the digits can be repeated?**
(b) **How many unique codes are possible if the digits cannot be repeated?**
(c) **In the case where digits may be repeated, how many codes are numbers that are greater than 2000 and even?**
(d) **In the case where digits may not be repeated, how many codes are numbers that are greater than 2000 and divisible by 4?**
(e) **What is the probability that a code will contain at least one 7? The digits may be repeated.**
(f) **What is the probability that a code will contain at least one 7? The digits may not be repeated.**
(g) **How many codes can be formed between 4000 and 5000? The digits may be repeated.**
**Explanation:** These questions involve combinatorial calculations with constraints on digit repetition and specific properties of the resulting codes (e.g., greater than 2000, even, divisible by 4). They aim to test understanding of permutations, combinations, and probabilities within a defined digit set.
**Educational Insight:** Understanding these principles can be fundamental in various areas of mathematical problem-solving and real-world applications like computing valid sequences for security systems or organizing unique identifiers in databases.
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