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 )

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. )

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

rosala
Feb 17, 2018

#1**+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....!!!!

CPhill
Feb 17, 2018

#2**+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

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

CPhill
Feb 17, 2018

#3**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!

rosala
Feb 19, 2018