Permutations and Combinations!

1.A telegraph has x arms and each arm is capable of (x-1) distinct positions , including the position of rest. The total no. of signals that can be made is?

Rosala, This is just      x(x-1)

2. How many natural numbers are there fro 1 to 1000 which have none of their digits repeated?

i've tried this question a no. of times but it just doesnt see to be clicked in my mind. i would be really appreciate if anyone can explain it to me in detail.

There are 9 one digit ones

There are no four digit ones

So how many 2 digit ones are there.

The tens digit can be 1 to 9 that is 9 choices, now you used one digit but you can use the zeros so there are 9 possible digits left for the units digit

So that is 9*9=81

AND how many three digit ones are then  9*9*8 = 648

Altogether there are  9+81+648 = 738

3. Number of different natural numbers which are smaller than two hundred million and using only the digits 1 or 2 is :

less than      200,000,000   only containing the digits 1 and 2

1 digit     2

2 digit     2*2=2^2

3 digit      2^3

4 digit       2^4

5 digit      2^5

6 digit      2^6

7 digit      2^7

8 digit      2^8

9 digit      2^8       The biggest value digit must be 1

So what do we get when we add these up

\(2+2^2+2^3+2^4+2^5+2^6+2^7+2^8+2^8\\ =(2+2^2+2^3+2^4+2^5+2^6+2^7+2^8)+2^8\\ \qquad \text{The brackets is the sum of a GP, a=2 and r=2\;\;n=8}\\ =\frac{a(r^n-1)}{r-1}\;\;\;+2^8\\ =\frac{2(2^8-1)}{2-1}\;\;\;+2^8\\ =2^9-2+2^8\\ =2^9+2^8-2\\ =2^8(2+1)-2\\ =3\times 2^8-2 \)

i. (3).2^8 - 2 

ii. (3).2^8 - 1 

iii. 2 (2^9 - 1) 

iv) None 

I have a few more question like the questions above, and i just dont seem to get the correct answers. 

i have tried the questions above many times but i just cant understand a few things , i would be glad if anyone could explain to me in detail. Thank you very much. 

i really appreciate the time you'd take out for my answers! 

Nicely explained, Melody....!!!!

thank you so much Melody for taking the time to answer my questions,

i'd like to start wiith the first one.

the answer that youve given to question no. 1 , thats not the answer thats present in my book. the answer to it is (x-1 )^x  - 1 , my doubt in this was that why are we subtracting one here? the (x-1)^x ca efro forula (n)^r.

2.oh Melody i feel bad to have troubled you for this problem , i was just making 2 silly mistakes. but couldnt see them,thank you for  the answer and explanation. 

3. 

#

less than      200,000,000   only containing the digits 1 and 2
1 digit     2

2 digit     2*2=2^2

3 digit      2^3

4 digit       2^4

5 digit      2^5

6 digit      2^6

7 digit      2^7

8 digit      2^8

9 digit      2^8       The biggest value digit must be 1

#

your third answer is correct btw!

i have no idea what you did here...otherwise i understand everything after that..... could you pls explain this part ? thank you for the answers! 

thank you so much Melody for taking the time to answer my questions,

i'd like to start wiith the first one.

the answer that youve given to question no. 1 , thats not the answer thats present in my book. the answer to it is (x-1 )^x  - 1 , my doubt in this was that why are we subtracting one here? the (x-1)^x ca efro forula (n)^r.

2.oh Melody i feel bad to have troubled you for this problem , i was just making 2 silly mistakes. but couldnt see them,thank you for  the answer and explanation. 

3. 

#

less than      200,000,000   only containing the digits 1 and 2
1 digit     2

2 digit     2*2=2^2

3 digit      2^3

4 digit       2^4

5 digit      2^5

6 digit      2^6

7 digit      2^7

8 digit      2^8

9 digit      2^8       The biggest value digit must be 1

#

your third answer is correct btw!

i have no idea what you did here...otherwise i understand everything after that..... could you pls explain this part ? thank you for the answers! 

1.A telegraph has x arms and each arm is capable of (x-1) distinct positions , including the position of rest. The total no. of signals that can be made is?

My answer      x(x-1)

Your given answer:      

 (x-1 )^x  - 1

Ok I was assuming that you chose one for each but it seems you can choose many more than that.

It must be more like question 3.

I really do not understand the question properly. Maybe I will think about it more later when I have more time. Can you explain what the question is better Rosala?

hi Melody, 

i just found a question which is similar to the 1st one and now i understand it. if you're interested, heres the answer....

can you pls reexplain the 3rd one tho?

Thanks for your help ! 

Hi Rosala,

Thanks for your feedback and for that answer.

Could you please try to explain what you don't understand about the 3rd one?

I am super busy over this time but I will try to get back to you. If not I do believe Chris understands perfectly and will be able to explain.

I hope I don't sound like I am taking you for granted Chris. It is just that you are here more than me and you are super generous with your time and knowledge:)