JEE Advanced · 2024 · 26 May 2024 · Paper 2 · Permutations and Combinations

**For $n(Y)$:** A relation with range having exactly one element means all 6 ordered pairs have the same second coordinate. But since $R$ must have 6 elements and there are only 6 elements in $S$, this would require all 6 first coordinates to be distinct. However, checking the constraint $|a-b|\ge 2$ for a fixed $b$, we cannot find 6 distinct values of $a$ satisfying this. Therefore, $n(Y)=0$.

**For $n(Z)$:** A function from $S$ to $S$ assigns exactly one $b$ to each $a$. From the earlier count:

• $a=1$: 4 choices
• $a=2$: 3 choices
• $a=3$: 3 choices
• $a=4$: 3 choices
• $a=5$: 3 choices
• $a=6$: 4 choices

Number of functions:


So $n(Y)+n(Z)=0+1296=1296=36^2=k^2$.

Therefore, $|k|=36$.