P&C

• wishboy

  16 May 10, 01:34

  Hi, I stumbled upon this question in my tutorial. It is taken from MJC/2009/P2/8.

  Find the number of different arrangements of the name "BENNETT TAN"

  (i) beginning with B and ending with N,                                          Ans: 1680
  (ii) with all the T's separated and all the N's separated.               Ans: 12120

  I got the answer for (i) by

  8!/(2!2!3!) = 1680

  but I'm not sure how to do (ii), how do I approach this question?

  Thanks in advance! :D

• wee_ws

  16 May 10, 08:06

  Hi,

    The steps are:

    1. Arrange A, B, E, E.

    2. Slot N, N, N in between A, B, E, E.

    3. Slot T, T, T in between the letters in step 2.

    The correct answer is 6720.

    Do take note that many P&C questions have the following problem-solving format:

    - Arrange the others.
    - Slot things in between so that they are separated.

    In the question we just saw, the process of slotting is carried out twice. Thanks!

  Cheers,
  Wen Shih

• wishboy

  16 May 10, 09:11

  Originally posted by wee_ws:

  Hi,

    The steps are:

    1. Arrange A, B, E, E.

    2. Slot N, N, N in between A, B, E, E.

    3. Slot T, T, T in between the letters in step 2.

    The correct answer is 6720.

    Do take note that many P&C questions have the following problem-solving format:

    - Arrange the others.
    - Slot things in between so that they are separated.

    In the question we just saw, the process of slotting is carried out twice. Thanks!

  Cheers,
  Wen Shih

  Yeah I thought about that initially, but couldn't get the answer provided.

  Consider this case,

  ABEETNTNTN (and other similar cases such as ABNTNTNTEE or ANTNTBENTE

  This is not attainable if I use the slotting in twice method, since I would be slotting more than 1 N into a slot.

• wee_ws

  16 May 10, 11:50

  Hi,

    You have a great point that I did not consider :P

    Back to the drawing board...

  Cheers,
  Wen Shih

• SBS261P

  16 May 10, 16:56

   

• Mad Hat

  16 May 10, 23:18

  The question is nasty. I can't find a simple way. Here's an ugly way. Anyone got a better way do share. Students should not be worried about this sort of question, which is sadistic!
  
  abee can be permuted in 12 ways

  Now add three Ts
  Case 1 ................. a b e e TTT ................. 5 ways
  Case 2 ................. a b e T e TT................. ⁵C₂ x 2 ways
  Case 3 ................. a b T e T e T................. ⁵C₃ ways
  
  Now add three Ns, the red ones being forced
  Case 1 ................. N a b e e TNTNT ................. 6 ways
  Case 2 ................. N a N b e T e TNT ................. ⁷C₂ ways
  Case 3 ................. N a N b N T e T e T................. ⁸C₃ ways
  
  Total
  (12 x 5 x 6) + (12 x ⁵C₂ x 2 x ⁷C₂) + (12 x ⁵C₃ x ⁸C₃) = 12,120

• wishboy

  17 May 10, 00:31

  Originally posted by Mad Hat:

  The question is nasty. I can't find a simple way. Here's an ugly way. Anyone got a better way do share. Students should not be worried about this sort of question, which is sadistic!
  
  abee can be permuted in 12 ways

  Now add three Ts
  Case 1 ................. a b e e TTT ................. 5 ways
  Case 2 ................. a b e T e TT................. ⁵C₂ x 2 ways
  Case 3 ................. a b T e T e T................. ⁵C₃ ways
  
  Now add three Ns, the red ones being forced
  Case 1 ................. N a b e e TNTNT ................. 6 ways
  Case 2 ................. N a N b e T e TNT ................. ⁷C₂ ways
  Case 3 ................. N a N b N T e T e T................. ⁸C₃ ways
  
  Total
  (12 x 5 x 6) + (12 x ⁵C₂ x 2 x ⁷C₂) + (12 x ⁵C₃ x ⁸C₃) = 12,120

  Wow thanks!

  Wondering if there is any simpler method to do this question...

• eagle

  17 May 10, 08:06

  Originally posted by wishboy:

  Yeah I thought about that initially, but couldn't get the answer provided.

  Consider this case,

  ABEETNTNTN (and other similar cases such as ABNTNTNTEE or ANTNTBENTE

  This is not attainable if I use the slotting in twice method, since I would be slotting more than 1 N into a slot.

  You are good! The solution provided by MJC itself says 6720.

  Looking at the solution, I guess this part wasn't on the examiner's mind when he/she set the paper.

  I think Mad Hat's method is very good and logical and easy to understand.

• wee_ws

  17 May 10, 08:51

  Hi Mad Hat,

    Thanks for sharing your solution which is a very good method. I agree with you that this question is highly difficult and is thus not suitable for A-level.

  Cheers,
  Wen Shih

   

• Mad Hat

  17 May 10, 16:00

  My first answer was 6720 also