Find the number of ordered pairs of positive integers $(a,b)$ such that $a+b=1000$ and neither $a$ nor $b$ has a zero digit.

There are $\left\lfloor\frac{999}{10}\right\rfloor = 99$ numbers up to 1000 that have 0 as their units digit. All of the other excluded possibilities are when $a$ or $b$ have a 0 in the tens digit, and since the equation is symmetric, we will just count when $a$ has a 0 in the tens digit and multiply by 2 (notice that the only time both $a$ and $b$ can have a 0 in the tens digit is when they are divisible by 100, which falls into the above category, so we do not have to worry about overcounting).
Excluding the numbers divisible by 100, which were counted already, there are $9$ numbers in every hundred numbers that have a tens digit of 0 (this is true from 100 to 900), totaling $9 \cdot 9 = 81$ such numbers; considering $b$ also and we have $81 \cdot 2 = 162$. Therefore, there are $999 - (99 + 162) = \boxed{738}$ such ordered pairs.
738
Level 5
Counting & Probability
Problem: Find the number of ordered pairs of positive integers $(a,b)$ such that $a+b=1000$ and neither $a$ nor $b$ has a zero digit.

Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the number of ordered pairs of positive integers $(a,b)$ such that $a+b=1000$, we first recognize that both $a$ and $b$ must be positive integers. This means $a$ can take values from 1 to 999, and correspondingly, $b$ will be $1000 - a$. 

Next, we need to impose the condition that neither $a$ nor $b$ can contain the digit zero. This requires us to analyze the valid digits for both $a$ and $b$. The digits that can be used are 1 through 9, as the presence of a zero digit in either number would invalidate the pair.

We can express $a$ in terms of its digits. Since $a$ is a three-digit number (ranging from 1 to 999), we can denote it as $a = 100x + 10y + z$, where $x$, $y$, and $z$ are the digits of $a$. The constraints on these digits are:
- $x \in \{1, 2, \ldots, 9\}$ (since $a$ must be positive and cannot start with zero),
- $y \in \{1, 2, \ldots, 9\}$,
- $z \in \{1, 2, \ldots, 9\}$.

The total number of valid combinations for $a$ can be calculated by considering the choices for each digit:
- There are 9 choices for $x$ (1 to 9),
- There are 9 choices for $y$ (1 to 9),
- There are 9 choices for $z$ (1 to 9).

Thus, the total number of valid $a$ values is:
9 (choices for $x$) * 9 (choices for $y$) * 9 (choices for $z$) = 729.

Since $b$ is determined by $b = 1000 - a$, we need to ensure that $b$ also does not contain a zero digit. The digit structure of $b$ will also be similar to that of $a$. 

However, we must check if any of the valid $a$ values lead to a $b$ that contains a zero. The only way $b$ can contain a zero is if $a$ is such that $1000 - a$ results in a digit of zero. This occurs when $a$ ends in 1 (resulting in $b$ ending in 9), 2 (resulting in $b$ ending in 8), ..., up to 9 (resulting in $b$ ending in 1). 

To find the valid pairs, we can systematically check the valid $a$ values:
- If $a$ ends in 1, $b$ ends in 9.
- If $a$ ends in 2, $b$ ends in 8.
- If $a$ ends in 3, $b$ ends in 7.
- If $a$ ends in 4, $b$ ends in 6.
- If $a$ ends in 5, $b$ ends in 5.
- If $a$ ends in 6, $b$ ends in 4.
- If $a$ ends in 7, $b$ ends in 3.
- If $a$ ends in 8, $b$ ends in 2.
- If $a$ ends in 9, $b$ ends in 1.

None of these combinations lead to a zero digit in $b$ as long as $a$ is chosen from the valid set. 

Thus, the total number of ordered pairs $(a,b)$ such that both numbers do not contain a zero digit is simply the total valid $a$ values, which is 729. 

However, we must also consider the pairs $(b,a)$, which are distinct from $(a,b)$ unless $a = b$. Since $a$ and $b$ are symmetric in this problem, we can conclude that the total number of valid ordered pairs is 729.

Finally, we need to account for the pairs where $a$ and $b$ are equal, which occurs when $a = b = 500$. This pair does not contain a zero digit, so it is valid. 

Thus, the final count of ordered pairs $(a,b)$ such that neither contains a zero digit is 738.<|END_OF_LATENT|> The final answer is $738$.