AIME 1983 · 第 10 题

Solution 1

Suppose that the two identical digits are both $1$. Since the thousands digit must be $1$, only one of the other three digits can be $1$. This means the possible forms for the number are

$11xy,\qquad 1x1y,\qquad1xy1$

Because the number must have exactly two identical digits, $x\neq y$, $x\neq1$, and $y\neq1$. Hence, there are $3\cdot9\cdot8=216$ numbers of this form.

Now suppose that the two identical digits are not $1$. Reasoning similarly to before, we have the following possibilities:

$1xxy,\qquad1xyx,\qquad1yxx.$

Again, $x\neq y$, $x\neq 1$, and $y\neq 1$. There are $3\cdot9\cdot8=216$ numbers of this form.

Thus the answer is $216+216=\boxed{432}$.

Solution 2

Consider all four-digit sequences with exactly two identical digits. By symmetry, $\frac{1}{10}$ of these sequences start with $1$.

Let $A$ be the digit that appears twice, and $B > C$ be the digits which appear once. Then there are $10$ choices for $A$ and $\binom{9}{2} = 36$ choices for $(B, C)$. Therefore, there are $10 \cdot 36 = 360$ choices for $(A, B, C)$.

For each choice of $(A, B, C)$, there are $\frac{4!}{2!} = 12$ permutations of $AABC$. Therefore, there are $12 \cdot 360 = 4320$ four-digit sequences with exactly two identical digits. Of these sequences, $\frac{1}{10} \cdot 4320 = \boxed{432}$ begin with $1$.

Solution 3 (Complementary Counting)

We'll use complementary counting. We will split up into $3$ cases: (1) no number is repeated, (2) $2$ numbers are repeated, and $2$ other numbers are repeated, (3) $3$ numbers are repeated, or (4) $4$ numbers are repeated.

Case 1: There are $9$ choices for the hundreds digit (it cannot be $1$), $8$ choices for the tens digit (it cannot be $1$ or what was chosen for the hundreds digit), and $7$ for the units digit. This is a total of $9\cdot8\cdot7=504$ numbers.

Case 2: One of the three numbers must be $1$, and the other two numbers must be the same number, but cannot be $1$ (That will be dealt with in case 4). There are $3$ choices to put the $1$, and there are $9$ choices (any number except for $1$) to pick the other number that is repeated, so a total of $3\cdot9=27$ numbers.

Case 3: We will split it into $2$ subcases: one where $1$ is repeated $3$ times, and one where another number is repeated $3$ times. When $1$ is repeated $3$ times, then one of the digits is not $1$. There are $9$ choices for that number, and $3$ choices for its location,so a total of $9\cdot3=27$ numbers. When a number other than $1$ is repeated $3$ times, then there are $9$ choices for the number, and you don't have any choices on where to put that number. So in Case 3 there are $27+9=36$ numbers

Case 4: There is only $1$ number: $1111$.

There are a total of $1000$ $4$-digit numbers that begin with $1$ (from $1000$ to $1999$), so by complementary counting you get $1000-(504+27+36+1)=\boxed{432}$ numbers.

Solution by Kinglogic

Solution 4 (Easy casework)

Let us proceed by casework.

Case 1: We will count the amount of numbers that have two identical digits that are not one. The thousands digit is fixed, and we are choosing two spots to hold two identical digits that are chosen from $0, 2-9$, which is $9$ options. For the last digit, their are $8$ possibilities since it can be neither $1$ or the other number that is chosen. The final outcome is ${3}\choose{2}$ $* 9 * 8 = 216$ possibilities for this case.

Case 2: The last case will be the amount of numbers that have two identical digits thare are $1$. There are ${3}\choose{1}$ places to pick the $1$. For the other $2$ digits, there are $9$ and $8$ options respectively. Thus, we have $3 * 9 * 8 = 216$. Summing the two cases, we get $216 + 216 = \boxed{432}$.