Counting problem social security numbers

[cragar]

Homework Statement

social security number is a 9 digit number.
the first digit may be 0
a. How many numbers are available
b. How many are even
c. How many have all of their digits even
d. How many read the same forward and backward
e. How many have none of their digits equal to 8
f. How many have at least one digit equal to 8
g. How many have exactly one 8.

The Attempt at a Solution

a. assuming we can have a number with all zero's i would say
$10^9$ because we have 10 possible choices for each of the 9 slots
b. I would assume half of part a
c. I would assume half unless there is some kind of fence post error.
d. I think it is 10*10*10*10*10*1*1*1*1
The middle number can be what-ever since its in the middle
so i have 10 choices for that. And the four numbers to the left of the middle can be anything so i have 10 choices for those but this restricts the numbers on the right so I have only 1 choice or those.
e. Since i have removed a possible choice I only have 9 choice for my 9 slots so it should be $9^9$
f. this should be $10^8$ because on one of them I only have one choice but the rest I have 10 choices.
g. On one of the slots I have one choice because it hast to be an 8 , but the rest can't have an 8 in them therefore I have 9 choices on the remaining 8 digits so the answer should be $9^8$

 

[Mark44]

Homework Statement

social security number is a 9 digit number.
the first digit may be 0
a. How many numbers are available
b. How many are even
c. How many have all of their digits even
d. How many read the same forward and backward
e. How many have none of their digits equal to 8
f. How many have at least one digit equal to 8
g. How many have exactly one 8.

The Attempt at a Solution

a. assuming we can have a number with all zero's i would say
$10^9$ because we have 10 possible choices for each of the 9 slots

Right.

b. I would assume half of part a

You should write this as a number.

c. I would assume half unless there is some kind of fence post error.

I don't think so. How many choices are there for each of the slots? Is this number half of 10⁹?

d. I think it is 10*10*10*10*10*1*1*1*1
The middle number can be what-ever since its in the middle
so i have 10 choices for that. And the four numbers to the left of the middle can be anything so i have 10 choices for those but this restricts the numbers on the right so I have only 1 choice or those.

Looks good, but you should simplify this number.

e. Since i have removed a possible choice I only have 9 choice for my 9 slots so it should be $9^9$

OK

f. this should be $10^8$ because on one of them I only have one choice but the rest I have 10 choices.

This one bothered me a little, but I think you're right.

g. On one of the slots I have one choice because it hast to be an 8 , but the rest can't have an 8 in them therefore I have 9 choices on the remaining 8 digits so the answer should be $9^8$

OK

 

[cragar]

thanks for your response.
and on part c. it should be $5^9$ because I have 5 even numbers to choose from on each slot.
and why does part f bother you.

 

[Mark44]

thanks for your response.
and on part c. it should be $5^9$ because I have 5 even numbers to choose from on each slot.

Which is quite a bit different from your first answer of .5 * 10^9

and why does part f bother you.

I was concerned that you were counting too many numbers.

 

[cragar]

ya i think i read part c to quickly . thanks for you help .

 

[cragar]

Now I think part f is wrong. Because all the ones that have at least one 8 should be the total numbers of possible social security numbers minus the ones that don't have any 8's
so i think the answer to f should be $10^9-9^9$
And now I am not so sure about g . Because if I had an 8 in the first slot I would have one choice then 9 choices for the following, Then I would have to do another chart with an 8 in the second slot and then keep going down the line.

 

[DaleSwanson]

Now I think part f is wrong. Because all the ones that have at least one 8 should be the total numbers of possible social security numbers minus the ones that don't have any 8's
so i think the answer to f should be $10^9-9^9$
And now I am not so sure about g . Because if I had an 8 in the first slot I would have one choice then 9 choices for the following, Then I would have to do another chart with an 8 in the second slot and then keep going down the line.

That's what I got for part f (612579511). For g I got the same thing as e (9⁹). At first this seemed wrong but I think it is right. As you said, 9⁸ is how many possibilities will have an 8 in (only) the first slot. All you have to do is recognize that any of the slots could be the one with the 8 and thus multiple that by the number of slots, so 9⁸ * 9 = 9⁹