# Permuations and Combinations Problem

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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mfb
Mentor
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, ...

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
Mentor
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?

mfb
Mentor
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 alot for the help!