Let and be distinct integers where and Then, the number of ways of choosing and such that is divisible by 5, is _________ . [2023]

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Solution

Q.
Let $x$ and $y$ be distinct integers where $1 \leq x \leq 25$ and $1 \leq y \leq 25 .$ Then, the number of ways of choosing $x$ and $y,$ such that $x + y$ is divisible by 5, is _________ . [2023]

Ans.
(120)

Let $x + y = 5 λ$

Possible cases are

$\begin{matrix} x & y & Number of ways \\ 5 λ & 5 λ & 20 \\ 5 λ + 1 & 5 λ + 4 & 25 \\ 5 λ + 2 & 5 λ + 3 & 25 \\ 5 λ + 3 & 5 λ + 2 & 25 \\ 5 λ + 4 & 5 λ + 1 & 25 \end{matrix}$

$Total number of ways = 20 + 25 + 25 + 25 + 25 = 120$

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