Permuations and Combinations Problem

[andyrk]

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!

 

[mfb]

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, ...

 

[andyrk]

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?

 

[mfb]

Which part is unclear?

 

[andyrk]

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?

 

[mfb]

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.

 

[andyrk]

Okay! Thanks a lot for the help!