# Thread: How many decimal n-tuples contain at least one each of {1,2,3}?

1. ## How many decimal n-tuples contain at least one each of {1,2,3}?

How many decimal n-tuples contain at least one each of {1,2,3}?

2. Originally Posted by qtpipi
How many decimal n-tuples contain at least one each of {1,2,3}?
There are a total of $\displaystyle 10^{n}$ decimal n-tuples.
How many of those contain none of $\displaystyle {1,2,3}$?

3. I'm not sure. Can you give another hint

4. ## Compute for the remainder

Originally Posted by qtpipi
I'm not sure. Can you give another hint
Decimal n-tuples are n-tuples of $\displaystyle \{1,2,3,4,6,7,8,9,0\}$ which has cardinality of 10.

From PLATO's hint on computing the number of distinct n-tuples from a set of cardinality 10:

Originally Posted by Plato
There are a total of $\displaystyle 10^{n}$ decimal n-tuples.
How many of those contain none of $\displaystyle 1,2,3$?
$\displaystyle \{1,2,3,4,5,6,7,8,9,0\}\setminus\{1,2,3\}=\{4,5,6, 7,8,9,0\}$

Use PLATO's hint to compute how many n-tuples can be made from the remainder set $\displaystyle \{4,5,6,7,8,9,0\}$ of cardinality 7. Then substract from the number of distinct n-tuples from a set of cardinality 10.

5. Originally Posted by qtpipi
How many decimal n-tuples contain at least one each of {1,2,3}?
qtpipi,

Could you please clarify the question?

Do you want the number of tuples that contain at least one each of all the digits 1,2,3-- for example, 93321-- or do you want the number of tuples that contain at least one 1, at least one 2, or at least one 3-- for example, 99111?

6. Originally Posted by awkward
Could you please clarify the question?
Do you want the number of tuples that contain at least one each of all the digits 1,2,3-- for example, 93321-- or do you want the number of tuples that contain at least one 1, at least one 2, or at least one 3-- for example, 99111?
Surely there is nothing to be clarified!
“At least one” is the complement of “none”.
$\displaystyle 10^{10}-7^{10}$.

7. Originally Posted by Plato
Surely there is nothing to be clarified!
“At least one” is the complement of “none”.
$\displaystyle 10^{10}-7^{10}$.
There is, in my mind, a difference between "at least one" and "at least one each of (some set)".

8. Originally Posted by awkward
There is, in my mind, a difference between "at least one" and "at least one each of (some set)".
That is fair enough. I should have read the question more carefully.
The number of decimals n-tuples not containing one of $\displaystyle 1,2,3$ is $\displaystyle 3\cdot 9^n - 3\cdot 8^n + 7^n$. WHY?

9. Originally Posted by Plato
That is fair enough. I should have read the question more carefully.
The number of decimals n-tuples not containing one of $\displaystyle 1,2,3$ is $\displaystyle 3\cdot 9^n - 3\cdot 8^n + 7^n$. WHY?
Neat!

I am sure there is more than one way to solve the problem. I would have said (looking at the original problem rather than the complementary question) that the number of decimal n-tuples which contain at least one each of 1, 2, and 3 is

$\displaystyle 10^n - 3 \cdot 9^n + 3 \cdot 8^n - 7^n$.

Why? Because that is the coefficient of $\displaystyle \frac{1}{n!} x^n$ in

$\displaystyle e^{7x}\;(e^x - 1)^3$.

(Exponential generating functions... gotta love them.)