How many three-digit numbers exist whose first and last digits add up to 9?

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:

  1. A=1,C=8A = 1, C = 8A=1,C=8
  2. A=2,C=7A = 2, C = 7A=2,C=7
  3. A=3,C=6A = 3, C = 6A=3,C=6
  4. A=4,C=5A = 4, C = 5A=4,C=5
  5. A=5,C=4A = 5, C = 4A=5,C=4
  6. A=6,C=3A = 6, C = 3A=6,C=3
  7. A=7,C=2A = 7, C = 2A=7,C=2
  8. A=8,C=1A = 8, C = 1A=8,C=1
  9. 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

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