Question about two different combination problems

[leo255]

1. Find the number of subsets of S = {1,2,3,...,10} that contain
   
   (a) the number 5.
   (b) neither 5 nor 6.
2. (c) both 5 and 6.
3. (d) no odd numbers.
4. e) exactly three elements.
5. (f) exactly three elements, all of them even.
6. (g) exactly five elements, including 3 or 4 but not both. (h) exactly five elements, but neither 3 nor 4.
7. (i) exactly five elements, the sum of which is even.

1. 2. Let A be the set of all strings of decimal digits of length seven. For example 0031227 and 1948301 are strings in A.
2. (a) Find |A|.
3. (b) How many strings in A begin with 1237 (in this order)?
4. (c) How many strings in A have exactly one 3?
5. (d) How many strings in A have exactly three 3s?
   

First of all, sorry about the numbers - the page seems to add numbers when I just want to get to a new line.

So, I'm trying to figure out the difference between what these two questions are asking (I am studying for a final next week).

I know that the answer to the first question is 10^9 and 10^7, respectively.

The answer to part b and c for the first question is 10^8. For c in the 2nd question, I would assume that the answer would be 10^6, but that is wrong. The answer is 7*(9^6). Also, for part d, the answer is C(7, 3) * 9^4, and I don't quite see how that is gotten.

Thanks for looking.

 

[mfb]

I know that the answer to the first question is 10^9 and 10^7, respectively.

10^9 is not the answer to any question, and I'm not sure what "first question" means as you have two problems with multiple subquestions. Mixing the two doesn't help.

The answer to part b and c for the first question is 10^8.

It is not.

Can you list some example subsets? I think you misinterpret question 1.

For c in the 2nd question, I would assume that the answer would be 10^6, but that is wrong.

Why do you assume this?

Also, for part d, the answer is C(7, 3) * 9^4, and I don't quite see how that is gotten.

The "3"s have to be somewhere. How many options do you have to fix their positions? What about the digits in the other 4 positions?

 

[haruspex]

I know that the answer to the first question is 10^9 and 10^7, respectively

I assume you are referring to 1a and 2a (see below). As mfb wrote, these are wrong. Did you make the same typo consistently?
Here's what I presume to be the right numbering:
1. Find the number of subsets of S = {1,2,3,...,10} that contain

(a) the number 5.
(b) neither 5 nor 6.
(c) both 5 and 6.
(d) no odd numbers.
(e) exactly three elements.
(f) exactly three elements, all of them even.
(g) exactly five elements, including 3 or 4 but not both.
(h) exactly five elements, but neither 3 nor 4.
(i) exactly five elements, the sum of which is even.​

2. Let A be the set of all strings of decimal digits of length seven. For example 0031227 and 1948301 are strings in A.

(a) Find |A|.
(b) How many strings in A begin with 1237 (in this order)?
(c) How many strings in A have exactly one 3?
(d) How many strings in A have exactly three 3s?​

For c in the 2nd question, I would assume that the answer would be 10^6, but that is wrong. The answer is 7*(9^6).

In how many places can the 3 appear?

 

[leo255]

@haruspex, thanks for cleaning up my numbering/lettering.

Yeah, my mistake - I meant b and c of #1 are 10^8.

Regarding 2 C, I think I see what is being asked. There are 7 different places the 3 could appear, and then after that, the rest of the 6 spots have nine available choices for numbers (everything besides the number 3). So, that would explain the 7 * 9^6. I also understand 2 D now.

 

[haruspex]

I meant b and c of #1 are 10^8

No, none of the answers are any power of 10. (Assuming that's 10 as in "ten", not some other base.)