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2004 AIME II Problems/Problem 4

Revision as of 20:10, 12 September 2020 by Jerry122805 (talk | contribs) (→‎Solution)

Problem

How many positive integers less than 10,000 have at most two different digits?

Solution

First, let's count numbers with only a single digit. We have nine of these for each length, and four lengths, so 36 total numbers.

Now, let's count those with two distinct digits. We handle the cases "0 included" and "0 not included" separately.

There are ${9 \choose 2}$ ways to choose two digits, $A$ and $B$. Given two digits, there are $2^n - 2$ ways to arrange them in an $n$-digit number, for a total of $(2^1 - 2) + (2^2 - 2) + (2^3 -2) + (2^4 - 2) = 22$ such numbers (or we can list them: $AB, BA, AAB, ABA, BAA, ABB, BAB, BBA, AAAB, AABA, ABAA,$ $BAAA, AABB, ABAB, BAAB, ABBA, BABA, BBAA, ABBB, BABB, BBAB, BBBA$). Thus, we have ${9 \choose 2} \cdot 22 = 36\cdot22 = 792$ numbers of this form.

Now, suppose 0 is one of our digits. We have nine choices for the other digit. For each choice, we have $2^{n - 1} - 1$ $n$-digit numbers we can form, for a total of $(2^0 - 1) + (2^1 - 1) + (2^2 - 1) + (2^3 - 1) = 11$ such numbers (or we can list them: $A0, A00, A0A, AA0, A000, AA00, A0A0, A00A, AAA0, AA0A, A0AA$). This gives us $9\cdot 11 = 99$ numbers of this form.

Thus, in total, we have $36 + 792 + 99 = \boxed{927}$ such numbers.


Solution 2

We use casework on the number of digits for this problem.

If the number has a single digit, namely the number $n \in [1,9],$ we can clearly all such $n$ work.

If the number has two digits, or the number $n \in [10,99]$ we can clearly see all such $n$ work.

If the number $n$ has three digits, there are a total of $900$ three digit numbers, and there are $9 \cdot 9 \cdot 8$ numbers that have all distinct digits so there are $900 - 9 \cdot 9 \cdot 8$ total three digit numbers that work.

If the number $n$ has four digits, then we have a total of $9 + \displaystyle\binom{9}{2}\left(\displaystyle\binom{4}{1}+ \displaystyle\binom{4}{2} + \displaystyle\binom{4}{3}\right) + 9 \cdot \left(\displaystyle\binom{3}{1}+\displaystyle\binom{3}{2} + \displaystyle\binom{3}{3}\right)$ so in total we get $\boxed{927}$ numbers that work.

See also

2004 AIME II (Problems • Answer Key • Resources)
Preceded by
Problem 3 		Followed by
Problem 5
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

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