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!