Permuations and Combinations Problem

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Discussion Overview

The discussion revolves around a permutations and combinations problem related to a shooting competition involving three targets with different numbers of rings. Participants are trying to understand the correct method to calculate the number of ways to hit the targets in a specified order.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • One participant seeks clarification on why the answer to the problem is option (C) and shares their understanding of the solution.
  • Another participant suggests disregarding the rings and focuses on the sequence of shots, proposing a sequence representation to analyze the problem.
  • There is a discussion about the number of ways to arrange the shots, with references to combinations such as C(12,4) for selecting positions for one of the targets.
  • Participants discuss the arrangement of shots and the formula for calculating the total arrangements, referencing factorials and combinations.
  • Clarifications are provided regarding the reasoning behind the use of combinations in the context of the problem.

Areas of Agreement / Disagreement

Participants express varying levels of understanding regarding the solution, with some agreeing on the use of combinations while others seek further clarification. The discussion does not reach a consensus on the best approach to the problem.

Contextual Notes

Some participants express confusion over specific steps in the reasoning, particularly regarding the application of combinations and the significance of the order of shots. There are unresolved aspects related to the interpretation of the problem's conditions.

andyrk
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Can someone please explain the following question? I am also posting the solution but I didn't understand it well enough. Why is the answer (C)? Can someone explain it to me?In a shooting competition, three targets are set as shown :

Condition :
Target (I) has four rings on which a person can hit in order from inside to outside.
Target (II) has three rings on which a person can hit in order from outside to inside.
Target (III) has five rings on which a person can hit in order from inside to outside.

The number of ways in which 12 shots (one at each ring) can be made :
[Hint : Any target can be chosen before not completing specific target but order of hit for a particular target should be as specified above in condition.]

(A) 12! /(4! 3! 5!)
(B) 12!/(4! 3! 5!) x 3!
(C) C(12,4) x C(8,3) x C(5,5)
(D) 12!

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Forget the rings for now - they get hit in the order of shots, who cares.

The shot sequence can be written like "ABCCBCBACAC" where A, B, C are the three disks. How many of those sequences are there? You have C(12,4) ways to pick positions for the "B"s, ...
 
mfb said:
Forget the rings for now - they get hit in the order of shots, who cares.

The shot sequence can be written like "ABCCBCBACAC" where A, B, C are the three disks. How many of those sequences are there? You have C(12,4) ways to pick positions for the "B"s, ...
Hey mfb, I didn't understand what you meant by the above statements. Could you please explain it more clearly?
 
Which part is unclear?
 
Why do we have C(12,4) ways to pick position for Bs?

Okay, I think I got this now. Suppose one of the possible sequence of shots is-
AAAABBBCCCCC (Suppose we complete the disk fully which we started). Now the number of ways to arrange this word is 12!/(4!x3!x5!) or C(12,4) (select 4 places from the 12 above) x C(8,3) (select 3 places from the 8 left above now) x C(5,5) (select 5 places from the 5 left now)

Is this the correct explanation?
 
andyrk said:
Why do we have C(12,4) ways to pick position for Bs?
You have 12 possible positions, and you have to choose 4.

Okay, I think I got this now. Suppose one of the possible sequence of shots is-
AAAABBBCCCCC (Suppose we complete the disk fully which we started). Now the number of ways to arrange this word is 12!/(4!x3!x5!) or C(12,4) (select 4 places from the 12 above) x C(8,3) (select 3 places from the 8 left above now) x C(5,5) (select 5 places from the 5 left now)

Is this the correct explanation?
Right.
 
Okay! Thanks a lot for the help!
 

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