Combinatroics 4-permuations of positive integers

In summary, there are 98 possible values for k and 5 locations for the alternate integer to be in the permutation. For part a), the number of 4-permutations containing three consecutive integers is 98*96*5 = 47040. However, the correct answer is 37927, suggesting that some permutations have been counted more than once. For part b), the number is 98*96*2 = 18816, which is closer to the book's answer of 18915. The reason for the discrepancy is that some permutations have been counted multiple times.
  • #1
farleyknight
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Homework Statement



How many 4-permutations of the positive integers not exceeding 100 contain three consecutive integers k, k+1, k+2, in the correct order:

a) where these consecutive integers can perhaps be separated by other integers in the permutation?

b) where they are in consecutive positions in the permutation?

Homework Equations



The Attempt at a Solution



I've already taken a look at the book's answers, but I don't seem how they arrived at them.

First off there can only be 98 possible values for k. Next, there is 5 locations for the alternate integer to be, and there are 96 integers left to choose from. So for part a) my guess was 98*96*5=47040. However, the book gives the answer: 37927, which isn't even divisible by 5..

For part b), I got a similar answer. 2 positions this time: 98*96*2 = 18816, but this time I was closer to the books: 18,915, which isn't divisible by 2.

My only guess is that I'm misreading the problem.. But I'm not sure what's the correct interpretation.
 
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  • #2
farleyknight said:
Next, there is 5 locations for the alternate integer to be, and there are 96 integers left to choose from.
This should be 4 and 97, which gets you closer to the book's answer.

But some permutations have been counted more than once. Can you see why?
 

1. What is combinatorics?

Combinatorics is a branch of mathematics that deals with counting and organizing objects in a systematic way. It involves studying patterns, structures, and relationships between different sets of objects.

2. What are permutations?

Permutations are the different ways that a set of objects can be arranged in a particular order. For example, the permutations of the letters "ABC" are ABC, ACB, BAC, BCA, CAB, and CBA.

3. What is the significance of 4-permutations of positive integers?

4-permutations of positive integers refer to the number of ways that four positive integers can be arranged in a particular order. This can be useful in various mathematical problems, such as determining the number of ways to arrange a deck of cards or the possible outcomes in a game.

4. How do you calculate 4-permutations of positive integers?

The formula for calculating 4-permutations of positive integers is nPr = n! / (n-r)!, where n represents the total number of objects and r represents the number of objects being selected for each permutation.

5. What are some real-life applications of 4-permutations of positive integers?

4-permutations of positive integers can be useful in various fields such as computer science, statistics, and engineering. It can be used to analyze and solve problems related to combinations, probability, and optimization in real-life situations. For example, it can be used to determine the number of possible outcomes in a lottery game or to find the number of ways to arrange items in a store display.

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