If two different numbers are taken from the set 0, 1, 2, 3, ..., 10; then the probability that their sum as well as absolute difference are both multiples of 4, is?

Let us take this case by case. First, take number0. The second numberphas to be chosen such thatp𕒴andp+0are divisible by4. Hence,pcan be either4or8.Now, take number1. We find that3+1is divisible by4but3𕒵is not. Similarly,5𕒵is divisible by4but5+1is not. You can see that for all odd numbers, no suchpexists, so only even numbers should be checked.First Number Second Number No. of cases04,8226,10240,8(However, case of0,4is already taken)162,10(However, case of2,6is already taken)180,4(However, both cases have been taken)0102,6(However, both cases have been taken)0Hence, total number of cases possible are2+2+1+1=6Total ways to choose2numbers from0,1,2,...,10are11C2=55Hence, probability=655