Let $N$ denote the number of $7$ digit positive integers have the property that their digits are in increasing order. Determine the remainder obtained when $N$ is divided by $1000$. (Repeated digits are allowed.)

Note that a $7$ digit increasing integer is determined once we select a set of $7$ digits. To determine the number of sets of $7$ digits, consider $9$ urns labeled $1,2,\cdots,9$ (note that $0$ is not a permissible digit); then we wish to drop $7$ balls into these urns. Using the ball-and-urn argument, having $9$ urns is equivalent to $8$ dividers, and there are ${8 + 7 \choose 7} = {15 \choose 7} = 6435 \equiv \boxed{435} \pmod{1000}$.
435
Level 5
Counting & Probability
Problem: Let $N$ denote the number of $7$ digit positive integers have the property that their digits are in increasing order. Determine the remainder obtained when $N$ is divided by $1000$. (Repeated digits are allowed.)

Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the number of 7-digit positive integers with digits in increasing order, we first recognize that the digits can range from 1 to 9. Since the digits must be in increasing order and can repeat, we can think of this as a problem of distributing 7 identical items (the digits) into 9 distinct boxes (the digits from 1 to 9).

We can use the "stars and bars" combinatorial method to solve this problem. In this method, the stars represent the digits we are choosing, and the bars represent the dividers between different digit values. 

Let \( x_i \) represent the number of times digit \( i \) (where \( i \) ranges from 1 to 9) appears in the number. We need to satisfy the equation:

\[ x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7 + x_8 + x_9 = 7 \]

where \( x_i \geq 0 \) for all \( i \). 

To apply the stars and bars theorem, we need to place 7 stars and 8 bars (since there are 9 digits, we need 8 dividers). The total number of symbols (stars + bars) is:

\[ 7 + 8 = 15 \]

The number of ways to arrange these symbols is given by the binomial coefficient:

\[ \binom{15}{8} \]

This coefficient counts the number of ways to choose 8 positions for the bars out of 15 total positions. 

Calculating \( \binom{15}{8} \):

\[
\binom{15}{8} = \frac{15!}{8! \cdot (15-8)!} = \frac{15!}{8! \cdot 7!}
\]

Calculating the factorials:

\[
15! = 15 \times 14 \times 13 \times 12 \times 11 \times 10 \times 9 \times 8!
\]

Thus,

\[
\binom{15}{8} = \frac{15 \times 14 \times 13 \times 12 \times 11 \times 10 \times 9}{7!}
\]

Calculating \( 7! = 5040 \):

Now, we compute the numerator:

\[
15 \times 14 = 210
\]
\[
210 \times 13 = 2730
\]
\[
2730 \times 12 = 32760
\]
\[
32760 \times 11 = 360360
\]
\[
360360 \times 10 = 3603600
\]
\[
3603600 \times 9 = 32432400
\]

Now, we divide by \( 5040 \):

\[
\frac{32432400}{5040} = 6435
\]

Thus, \( N = 6435 \). 

Finally, we need to find the remainder when \( N \) is divided by 1000:

\[
6435 \mod 1000 = 435
\]

The final answer is 435.<|END_OF_LATENT|> The final answer is $435$.