To find the number of 3-digit numbers where the sum of the digits is equal to 25, let's proceed step-by-step:
1. Identify the range of 3-digit numbers: A 3-digit number ranges from 100 to 999.
2. Represent a 3-digit number: Let's represent a 3-digit number as [tex]\( \overline{abc} \)[/tex] where [tex]\( a, b, \)[/tex] and [tex]\( c \)[/tex] are the digits.
3. Condition for the sum: We need the sum of the digits [tex]\( a + b + c \)[/tex] to equal 25.
4. Possible values for [tex]\( a \)[/tex]: Since [tex]\( a \)[/tex] is the hundreds digit (and must be between 1 and 9 for it to be a 3-digit number), we can identify the possible values for [tex]\( b \)[/tex] and [tex]\( c \)[/tex] accordingly.
- If [tex]\( a = 9 \)[/tex], then [tex]\( b + c = 16 \)[/tex]
- If [tex]\( a = 8 \)[/tex], then [tex]\( b + c = 17 \)[/tex]
- If [tex]\( a = 7 \)[/tex], then [tex]\( b + c = 18 \)[/tex]
- If [tex]\( a = 6 \)[/tex], then [tex]\( b + c = 19 \)[/tex]
- If [tex]\( a = 5 \)[/tex], then [tex]\( b + c = 20 \)[/tex]
5. Finding valid combinations:
- For [tex]\( a = 9 \)[/tex]:
- [tex]\( b + c = 16 \)[/tex]
- Possible pairs: (9, 7), (8, 8), (7, 9) - 3 pairs.
- For [tex]\( a = 8 \)[/tex]:
- [tex]\( b + c = 17 \)[/tex]
- Possible pairs: (9, 8), (8, 9) - 2 pairs.
- For [tex]\( a = 7 \)[/tex]:
- [tex]\( b + c = 18 \)[/tex]
- Possible pair: (9, 9) - 1 pair.
6. Summarize the results:
- For [tex]\( a = 9 \)[/tex]: 3 valid combinations.
- For [tex]\( a = 8 \)[/tex]: 2 valid combinations.
- For [tex]\( a = 7 \)[/tex]: 1 valid combination.
Finally, adding all these valid combinations, we get the total count:
[tex]\[ 3 + 2 + 1 = 6 \][/tex]
So, there are 6 three-digit numbers for which the sum of the digits equals 25.
