Solution: Counting Three-Digit Numbers

The problem asks for the total number of three-digit numbers of the form "abc" where the digits $a$, $b$, and $c$ are all different, and the middle digit $b$ is the sum of the other two digits ($b = a + c$).

Constraints Analysis

• The number must be a three-digit number, so the first digit $a$ cannot be 0. Thus, $a$ must be from the set $\{1, 2, ..., 9\}$.
• Digits $b$ and $c$ can be any digit from 0 to 9.
• All digits must be different: $a &neq b$, $a &neq c$, and $b &neq c$.
• The condition relating the digits is $b = a + c$.
• Since $b$ must be a single digit (i.e., $b ≤ 9$), the sum $a + c$ must also be less than or equal to 9.

Calculating Valid Numbers by First Digit 'a'

We can find the total count by iterating through possible values for the first digit $a$ (from 1 to 9). For each $a$, we determine the possible values for $c$ such that $a + c ≤ 9$ and all digits $a$, $b = a + c$, and $c$ are distinct.

• If $a = 1$: We need $c$ such that $1 + c ≤ 9$ and $1, 1+c, c$ are distinct. Possible $c$ values range from 0 to 8. - If $c=0$, $b=1$. Digits are (1, 1, 0). Not distinct ($a=b$). - If $c=1$, $b=2$. Digits are (1, 2, 1). Not distinct ($a=c$). - Valid $c$ values are 2, 3, 4, 5, 6, 7, 8. (7 numbers: 132, 143, 154, 165, 176, 187, 198). Count for $a=1$ is 7.
• If $a = 2$: We need $c$ such that $2 + c ≤ 9$ and $2, 2+c, c$ are distinct. Possible $c$ values range from 0 to 7. - If $c=0$, $b=2$. Digits are (2, 2, 0). Not distinct ($a=b$). - If $c=2$, $b=4$. Digits are (2, 4, 2). Not distinct ($a=c$). - Valid $c$ values are 1, 3, 4, 5, 6, 7. (6 numbers: 231, 253, 264, 275, 286, 297). Count for $a=2$ is 6.
• If $a = 3$: We need $c$ such that $3 + c ≤ 9$ and $3, 3+c, c$ are distinct. Possible $c$ values range from 0 to 6. - If $c=0$, $b=3$. Digits are (3, 3, 0). Not distinct ($a=b$). - If $c=3$, $b=6$. Digits are (3, 6, 3). Not distinct ($a=c$). - Valid $c$ values are 1, 2, 4, 5, 6. (5 numbers: 341, 352, 374, 385, 396). Count for $a=3$ is 5.
• If $a = 4$: We need $c$ such that $4 + c ≤ 9$ and $4, 4+c, c$ are distinct. Possible $c$ values range from 0 to 5. - If $c=0$, $b=4$. Digits are (4, 4, 0). Not distinct ($a=b$). - If $c=4$, $b=8$. Digits are (4, 8, 4). Not distinct ($a=c$). - Valid $c$ values are 1, 2, 3, 5. (4 numbers: 451, 462, 473, 495). Count for $a=4$ is 4.
• If $a = 5$: We need $c$ such that $5 + c ≤ 9$ and $5, 5+c, c$ are distinct. Possible $c$ values range from 0 to 4. - If $c=0$, $b=5$. Digits are (5, 5, 0). Not distinct ($a=b$). - Valid $c$ values are 1, 2, 3, 4. (4 numbers: 561, 572, 583, 594). Count for $a=5$ is 4.
• If $a = 6$: We need $c$ such that $6 + c ≤ 9$ and $6, 6+c, c$ are distinct. Possible $c$ values range from 0 to 3. - If $c=0$, $b=6$. Digits are (6, 6, 0). Not distinct ($a=b$). - Valid $c$ values are 1, 2, 3. (3 numbers: 671, 682, 693). Count for $a=6$ is 3.
• If $a = 7$: We need $c$ such that $7 + c ≤ 9$ and $7, 7+c, c$ are distinct. Possible $c$ values range from 0 to 2. - If $c=0$, $b=7$. Digits are (7, 7, 0). Not distinct ($a=b$). - Valid $c$ values are 1, 2. (2 numbers: 781, 792). Count for $a=7$ is 2.
• If $a = 8$: We need $c$ such that $8 + c ≤ 9$ and $8, 8+c, c$ are distinct. Possible $c$ values are 0 and 1. - If $c=0$, $b=8$. Digits are (8, 8, 0). Not distinct ($a=b$). - Valid $c$ is 1. (1 number: 891). Count for $a=8$ is 1.
• If $a = 9$: We need $c$ such that $9 + c ≤ 9$. The only possibility is $c=0$. - If $c=0$, $b=9$. Digits are (9, 9, 0). Not distinct ($a=b$). Count for $a=9$ is 0.

Total Calculation

Summing the counts for each possible value of $a$: $ 7 + 6 + 5 + 4 + 4 + 3 + 2 + 1 + 0 = 32 $ Therefore, there are 32 such possible three-digit numbers.