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Please answer correctly. Deals with permutations and combinations

Please answer correctly. Deals with permutations and combinations

A First Course in Probability (10th Edition)
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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Please answer correctly. Deals with permutations and combinations

**Question:** How many numbers between 1 and 100 (inclusive) are divisible by 5 or 3? **Answer Field:** (An empty space is provided for the user to input their answer.) --- **Explanation:** To solve this problem, the goal is to count all numbers in the specified range that are divisible by either 5 or 3. 1. **Divisibility by 5:** - Numbers divisible by 5 between 1 and 100 are 5, 10, 15, ..., 100. - These numbers form an arithmetic sequence with the first term 5, last term 100, and common difference 5. 2. **Divisibility by 3:** - Numbers divisible by 3 between 1 and 100 are 3, 6, 9, ..., 99. - These numbers form an arithmetic sequence with the first term 3, last term 99, and common difference 3. 3. **Divisibility by both 5 and 3 (i.e., 15):** - Numbers divisible by 15 between 1 and 100 are 15, 30, 45, ..., 90. - These numbers form an arithmetic sequence with the first term 15, last term 90, and common difference 15. By applying the principle of inclusion-exclusion: - Count numbers divisible by 5 [\( n_5 \)] - Count numbers divisible by 3 [\( n_3 \)] - Subtract numbers divisible by both to avoid double counting [\( n_{15} \)] Total numbers = \( n_5 + n_3 - n_{15} \)
Transcribed Image Text:**Question:** How many numbers between 1 and 100 (inclusive) are divisible by 5 or 3? **Answer Field:** (An empty space is provided for the user to input their answer.) --- **Explanation:** To solve this problem, the goal is to count all numbers in the specified range that are divisible by either 5 or 3. 1. **Divisibility by 5:** - Numbers divisible by 5 between 1 and 100 are 5, 10, 15, ..., 100. - These numbers form an arithmetic sequence with the first term 5, last term 100, and common difference 5. 2. **Divisibility by 3:** - Numbers divisible by 3 between 1 and 100 are 3, 6, 9, ..., 99. - These numbers form an arithmetic sequence with the first term 3, last term 99, and common difference 3. 3. **Divisibility by both 5 and 3 (i.e., 15):** - Numbers divisible by 15 between 1 and 100 are 15, 30, 45, ..., 90. - These numbers form an arithmetic sequence with the first term 15, last term 90, and common difference 15. By applying the principle of inclusion-exclusion: - Count numbers divisible by 5 [\( n_5 \)] - Count numbers divisible by 3 [\( n_3 \)] - Subtract numbers divisible by both to avoid double counting [\( n_{15} \)] Total numbers = \( n_5 + n_3 - n_{15} \)
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