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avatar+11847 

1. Define a "good word" as a sequence of letters that consists only of the letters A, B, and C and in which A never immediately followed by B , B is never immediately followed by C , and C is never immediately followed byA. If the nuber of n-letter good words are 384, find the value of n.

(i have tried to try this question but my ind goes blank whenever i read it, i have no idea what "good word " is or how in the world can i even use PnC to solve this problem. if it wasnt in the chapter, i'd never know it was of this topic. lol cheeky)

 

2. There are 21 balls which are either white or black and the ball of the same colour are alike. Find the no. of white balls so that the number of arrangements of these balls in a row be maximum.

 

(i dont know what "maximum" means here . and i also dont know how to solve this question. cheeky)

 

3. How any 4 digit numbers are there which are divible by 2.

(my answer here is coming as 9*8*7*5 but the book's answer is 9*10*10*5... i dont know why im telling you this becoz then you can twist your answer to match the books, lol.)

 

 

my previous questions and doubts if you're interested!

 

https://web2.0calc.com/questions/permutations-and-combinations_9

 

and just a reminder to all those helpful matheaticians who take out their time to help us and make our concepts clearer. we really appreciate it. Thank you so much.

 

https://web2.0calc.com/questions/a-shoutout-post-for-cphill-and-all-those-helping-mathematicians

 

PnC seems to be really twisting my brain. its literally lowering my self esteem so badly, you have no idea.lol

 

cheekyindecision

rosala  Feb 17, 2018
edited by rosala  Feb 17, 2018
edited by rosala  Feb 17, 2018
 #1
avatar+87323 
+2

Number 3

 

Rosala....here's the explanation for the book's answer :

 

Note...if fthe leading digit cannot be 0, we have 9 choices for this

 

And for the next two digits, we have 10 ways to choose each

 

But note....that  the last digit must be even if the number is divisible by 2....so....we only have 5 choices for this  2,4,6, 8 or 0

 

So..... we have    9 * 10 * 10 *  5  =    4500 

 

Hope that helps....!!!!

 

 

cool cool cool

CPhill  Feb 17, 2018
 #2
avatar+87323 
+2

1)  Rosala....I  believe  the value of  n  is 8

 

Reasoning

 

The first letter can be chosen 3 ways...either A, B or C

 

But....the second letter can only be chosen 2 ways.....for instance....if A  is the first letter, the second can only be chosen  2  ways, either another  A  or   C

 

And notice that each successive letter can only be chosen in 2 ways......for instance...if the second letter is B, the third letter can only be another B  or A

 

So.....we need to solve this  to find the number of additional letters, m, after the first one

 

 3 * 2^m  =  384       divide by 3

 

2^m  = 128

 

And m  =  7  makes this true

 

So....the the number of total letters  is

 

First one + m additional    =  1  + 7  =  8

 

Another way to see this is

 

3 * 2 * 2 * 2 * 2 * 2 * 2 * 2  =  384

 

 

cool cool cool

CPhill  Feb 17, 2018
edited by CPhill  Feb 17, 2018
 #3
avatar+11847 
0

CPhill, in 3rd one, why have you repeated "10" , lets see it like this..

 

__ __ __ __

 

now we have 9 choices for the 1000th digit. (1,2,3,4,5,6,7,8,9)

 

next, after we have 5 choices for the ones place digit ( 0,2,4,6,8)

 

so now for the 100th place, we only have left 8 choices to choose from.

 

i thought that to made a no. divisible by 2 , only the last digit has to be even..? is that not true?

 

so that way we'll have 7 choices for the 10th position...

 

if we do it by 9*10*10*5

 

arent we repeating the digits?

 

 

_________________________________________________

 

And CPhill in the 1st one arent you repeating the letters? what does "good word " mean?

this is getting really confusing, can you pls give me a more detailed answer?

 

Don't you know how to do the 2nd question?

 

thank you so much for your help!

 

smiley

rosala  Feb 19, 2018

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