Sure! Here's the transcription for the educational website:---**Problem 4: The Greater Knightrola Small Monster Association**The Greater Knightrola Small Monster Association (GKSMA) has four chartered positions: President, Vice President, Treasurer, and Secretary. Additionally, it has 12 officers chosen at large. All of these seats must be filled by members of GKSMA, of which there are 50 members in total. No member may hold more than one position.**Question:** How many different ways are there to fill the positions? *Note:* Two ways are considered different if a person holds a different role or no role in the other way. For instance, if Alice is President in one arrangement and Vice President in another, those two arrangements must be counted differently. However, if she is an at-large member in both, that alone does not differentiate the arrangements.---

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Description: Sure! Here's the transcription for the educational website:---**Problem 4: The Greater Knightrola Small Monster Association**The Greater Knightrola Small Monster Association (GKSMA) has four chartered positions: President, Vice President, Treasurer,

Introduction: According to the permutation rule, if r distinct items are to be chosen from n items without replacement, in such a way that the order of selection matters, then the selection can be made in (nPr) = n! / (n – r)! = n (n – 1) … (n – r + 1) ways. According to the combination rule, if r distinct items are to be chosen from n items without replacement, in such a way that the order of selection does not matter, then the selection can be made in (nCr) = n! / [r! (n – r)!] = [n (n – 1) … (n – r + 1)] / r! ways.Calculation: Among the 4 chartered positions, the position of an individual is important. As a result, the order of selection is important while filling these 4 positions. Now, 4 people can be chosen from among the 50 people, one-by-one, to fill these 4 positions in the following number of ways: (50P4) = (50!) / (50 – 4)! = 50 × 49 × 48 × 47 = 5,527,200. For each permutation above, 12 officers can be selected from among the remaining 46 [= 50 – 4] people in the following number of ways: (46C12) = (46!) / [(12!) × (46 – 12)!] = (46!) / [(12!) × (34)!] = 38,910,617,655. Thus, the total number of ways to fill the positions is: (50P4) × (46C12), that is, (5,527,200 × 38,910,617,655).

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4. The Greater Knightrola Small Monster Association has four chartered positions: President, Vice President, Treasurer and Secretary; it also has 12 officers chosen at large. All of these seats must be filled by members of GKSMA, of which there are 50. No member may hold more than 1 position. How many different ways are there to fill the positions? (Note: Two ways are considered different if there is a person in one way who has a different role or no role in the other way. For example, if Alice is President in one way and she is Vice President in another, those two ways must count differently. But if she is an at large member in both ways, that is not by itself enough for those ways to be different.

4. The Greater Knightrola Small Monster Association has four chartered positions: President, Vice President, Treasurer and Secretary; it also has 12 officers chosen at large. All of these seats must be filled by members of GKSMA, of which there are 50. No member may hold more than 1 position. How many different ways are there to fill the positions? (Note: Two ways are considered different if there is a person in one way who has a different role or no role in the other way. For example, if Alice is President in one way and she is Vice President in another, those two ways must count differently. But if she is an at large member in both ways, that is not by itself enough for those ways to be different.

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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Concept explainers

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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**Problem 4: The Greater Knightrola Small Monster Association**

The Greater Knightrola Small Monster Association (GKSMA) has four chartered positions: President, Vice President, Treasurer, and Secretary. Additionally, it has 12 officers chosen at large. All of these seats must be filled by members of GKSMA, of which there are 50 members in total. No member may hold more than one position.

**Question:**
How many different ways are there to fill the positions?

*Note:* Two ways are considered different if a person holds a different role or no role in the other way. For instance, if Alice is President in one arrangement and Vice President in another, those two arrangements must be counted differently. However, if she is an at-large member in both, that alone does not differentiate the arrangements.

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