UGC NET CSE | December 2012 | Part 3 | Question: 25

Question

The number of distinct bracelets of five beads made up of red, blue and green beads (two bracelets are indistinguishable if the rotation of one yield another) is,

• A. 243
• B. 81
• C. 51
• D. 47

Answer 1

Combination of 5 beads same color =RRRRR,BBBBB,GGGGG

4 same colors = RRRRB ,RRRRG (similarly Blue and green colors ) =(5!/4! +5!/4!) ⨉3

3 same colors =RRRGB, RRRGG,RRRBB (Similarly other colors)=(5!/3! + 5!/3!2! + 5!/3!2!)⨉3

2 -2 same color , RRBBG, RRGGB, BBGGR =5!/2!2! ⨉3

Adding all we get answer 273

Comments

but ans given is 51

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Can u please explain in more detail

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You are wrong

Answer 2

Verify that any necklace may be characterized via one of the patterns below.

Recall that there are 3!/(3−n)! ways to put 3 things into n ordered slots.

xxxxx: 3 = 3!/(3−1)!
xxxxy: 6 = 3!/(3−2)!
xxxyy: 6 = 3!/(3−2)!
xxxyz: 6 = 3!/(3−3)!
xxyyz: 6 = 3!/(3−3)!
xxyxy: 6 = 3!/(3−2)!
xxyxz: 6 = 3!/(3−3)!
xxyzy: 6 = 3!/(3−3)!
xyzyz: 6 = 3!/(3−2)!
3 + 6 + 6 + 6 + 6 + 6 + 6 + 6 + 6 = 51

This is from @stack overflow. Can anybody explains it little bit more.