Consider numbers of the form $1a1$, where $a$ is a digit. How many pairs of such numbers are there such that their sum is also a palindrome?
Note: A palindrome is a number which reads the same from left to right and from right to left. Examples: $353$, $91719$.
Notice, that for two choices of $a$, the sum of the two $1a1$'s will be a palindrome iff the sum of the two $a$'s, call it $n$, is less than $10$. ExplanationIf this is the case, then the sum of the two numbers of the form $1a1$ would be $2n2$, a palindrome. However, if the sum is greater than or equal to $10$, it will be a two digit integer, say $1n$ (where $1$ is the tens digit and $n$ the units digit). Thus, the sum of the two numbers of the form $1a1$ would be $3n2$, which can't be a palindrome.. The number of ways to choose two numbers from ${0,1,2,3,4,5,6,7,8,9}$ with replacement such that their sum is less than $10$ is $10+8+6+4+2=30$. Thus, there are $\boxed{30}$.