TIFR CSE 2024 | Part A | Question: 4

Question

Let $z_{1}, z_{2}, z_{3}, \ldots, z_{2023}$ be a permutation of the numbers $1,2,3, \ldots, 2023$. Which of the following is true about the product $\prod_{i=1}^{2023}\left(z_{i}-i\right)$ ?

Note: The parity of an integer $n$ just denotes whether $n$ is even or odd. Formally, the parity of $n$ is said to be odd if $n$ is odd, and even if $n$ is even.

• A. The above product is always even.
• B. The above product is always odd.
• C. The parity of the above product always changes if we swap the values of any two variables among $z_{1}, z_{2}, \ldots, z_{2023}$.
• D. There always exist two variables among $z_{1}, z_{2}, \ldots, z_{2023}$ such that the parity of the above product changes if we swap their values, but there may also exist two variables among the $z_{1}, z_{2}, \ldots, z_{2023}$ such that swapping their values does not change the parity of the above product.
• E. None of the above statements is true.

   

Answer 1

We must check the parity of i and the \sigma(i). The only possible way to get the product odd is when both the parities don't match. For such a permutation to exist, there must be an equal number of odd and even numbers. But we have the set of numbers as {1,2...2023}, which means we have more odd numbers than even. Therefore, such a permutation cannot exist. Therefore, the product can never be odd, hence always even.