To find how many three-digit numbers exist where the first and last digits add up to 9, let’s follow a structured approach.
Step 1: Definition of a Three-Digit Number
A three-digit number can be written in the form ABCABCABC, where:
- AAA is the first digit (hundreds place).
- BBB is the second digit (tens place).
- CCC is the third digit (units place).
The first digit AAA must be between 1 and 9 (non-zero for a three-digit number), and CCC must satisfy A+C=9A + C = 9A+C=9.
Step 2: Possible Pairs of AAA and CCC
Given A+C=9A + C = 9A+C=9, we can list all valid pairs of AAA and CCC:
- A=1,C=8A = 1, C = 8A=1,C=8
- A=2,C=7A = 2, C = 7A=2,C=7
- A=3,C=6A = 3, C = 6A=3,C=6
- A=4,C=5A = 4, C = 5A=4,C=5
- A=5,C=4A = 5, C = 4A=5,C=4
- A=6,C=3A = 6, C = 3A=6,C=3
- A=7,C=2A = 7, C = 2A=7,C=2
- A=8,C=1A = 8, C = 1A=8,C=1
- A=9,C=0A = 9, C = 0A=9,C=0
This gives 9 valid pairs for AAA and CCC.
Step 3: Choosing the Middle Digit BBB
The middle digit BBB can be any digit from 0 to 9, since there are no restrictions on BBB.
This provides 101010 choices for BBB.
Step 4: Total Number of Valid Numbers
For each of the 9 valid AAA-CCC pairs, there are 101010 choices for BBB. Therefore, the total number of valid three-digit numbers is:9×10=909 \times 10 = 909×10=90
Final Answer:
There are 90 three-digit numbers whose first and last digits add up to 9.
Related:- Summation of Even Natural Numbers