Combinatorics: Choosing Books on a Shelf

In summary: So the total is 2412, as you said.In summary, there are 2412 ways to make a row of three books in which exactly one language is missing, given nine different English books, seven different French books, and five different German books. This is calculated by breaking the problem into six cases and using the combination formula for each case.
  • #1
Shoney45
68
0

Homework Statement


Given nine different English books, seven different French books, and five different German books: How many ways are there to mak a row of three books in which exactly one language is missing?




Homework Equations



P(n,k) C(n,k)

The Attempt at a Solution



I broke this up into six cases: Let English books be represented by E, German books by G, and French books by F. My six cases then are:

(2E)F = p(9,2)*7 = 504
(2E)G = p(9,2)*5 = 360
(2F)E = p(7,2)*9 = 378
(2F)G = p(7,2)*5 = 210
(2G)E = p(5,2)*9 = 540
(2G)F = p(5,2)*7 = 420

all of which equals 2412 possibilities.
 
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  • #2
Hi Shoney45! :smile:
Shoney45 said:
Given nine different English books, seven different French books, and five different German books: How many ways are there to mak a row of three books in which exactly one language is missing?

(2E)F = p(9,2)*7 = 504
(2E)G = p(9,2)*5 = 360
(2F)E = p(7,2)*9 = 378
(2F)G = p(7,2)*5 = 210
(2G)E = p(5,2)*9 = 540
(2G)F = p(5,2)*7 = 420

The order of the books doesn't matter, so it's not p. :wink:

(and where did your 60 come from in the last two? :confused:)
 
  • #3
tiny-tim said:
Hi Shoney45! :smile:


The order of the books doesn't matter, so it's not p. :wink:

(and where did your 60 come from in the last two? :confused:)

Sorry, but I don't understand what you mean by the 60 in my last two.

Never mind. I just figured out what you meant. That was just bad arithmetic.
 

1. What is combinatorics?

Combinatorics is a branch of mathematics that deals with counting, arranging, and selecting objects or elements in a specific order.

2. What is the purpose of studying combinatorics?

The purpose of studying combinatorics is to understand and solve problems related to counting and arranging objects in a systematic way, which has applications in various fields such as computer science, engineering, and statistics.

3. What is the "Choosing Books on a Shelf" problem in combinatorics?

The "Choosing Books on a Shelf" problem in combinatorics involves finding the number of ways to arrange a set of books on a shelf, considering the order of the books is important.

4. How do you approach the "Choosing Books on a Shelf" problem?

To solve the "Choosing Books on a Shelf" problem, you can use the fundamental principle of counting, which states that the total number of ways to perform a sequence of tasks is equal to the product of the number of ways to perform each task individually. In this case, you would multiply the number of books by the number of ways to arrange them.

5. Can the "Choosing Books on a Shelf" problem be generalized to other scenarios?

Yes, the "Choosing Books on a Shelf" problem can be generalized to other scenarios, such as arranging a set of objects in a particular order or selecting a subset of objects from a larger set. The principles of combinatorics can be applied to solve these types of problems as well.

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