Question

Number of $5$ digits even palindromes is:
A) $400$
B) $500$
C) $4000$
D) $5000$

Answer 1

A palindrome is a number that reads the same forward and backward, such as $242$. The five digit palindrome is of the form of $ABCBA$; where $A$ is the same digit on $1^{st}$ and $5^{th}$ place, $B$ is the same digit on $2^{nd}$ and $4^{th}$ place, $C$ is the digit on $3^{rd}$ place.

Complete step-by-step answer:
Let the five digit palindrome is of the form of $ABCBA$; where $A$ is the same digit on $1^{st}$ and $5^{th}$ place, $B$ is the same digit on $2^{nd}$ and $4^{th}$ place, $C$ is the digit on $3^{rd}$ place.
The palindrome must be even, if the number in position $A$ can only be $2,4,6$ or $8$. This number cannot be $0$ because if the first digit is $0$, it would make the $5$ digit number a $4$ digit number. So, the place $A$ can be filled by $4$ ways.
The number in position $B$could be $0,1,2,3,4,5,6,7,8,9$. So the place $B$ can be filled by $10$ ways.
Also, the number in position $C$ could be the same, i.e., $0,1,2,3,4,5,6,7,8,9$. So the place $C$ can be filled by $10$ ways.
So, the desired number of $5$ digits even palindromes$= 4 \times 10 \times 10$ $= 400$

Option A is the correct answer.

Note: An another method to solve this problem is described as follows:
The first and last digits can only be even integers and are the same in $4$ ways, i.e., $2,4,6$ or $8$.
The second and fourth digits are the same but ranges from $0 - 9$ in $10$ ways.
The third digit could be any integer from $0 - 9$ in $10$ ways.
So the desired number of $5$ digits even palindromes $= 4 \times 10 \times 10 = 400$