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  4. The number of ways of selecting two numbers a and b, a ∈ 2,4,6, ldots ., 100 and b ∈ 1,3,5, ldots . ., 99 such that 2 is the remainder when a+b is divided by 23 is

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Q. The number of ways of selecting two numbers $a$ and $b, a \in\{2,4,6, \ldots ., 100\}$ and $b \in\{1,3,5, \ldots . ., 99\}$ such that $2$ is the remainder when $a+b$ is divided by $23$ is

JEE MainJEE Main 2023Permutations and Combinations

Solution:

$a \in\{2,4,6,8,10, \ldots ., 100\} $
$b \in\{1,3,5,7,9, \ldots \ldots, 99\}$
Now, $a+b \in\{25,71,117,163\}$
(i) $a+b=25$, no. of ordered pairs (a, b) is 12
(ii) $a + b =71$, no. of ordered pairs (a, b) is 35
(iii) $a + b =117$, no. of ordered pairs (a, b) is 42
(iv) $a+b=163$, no. of ordered pairs (a, b) is 19
$\therefore$ total $=108$ pairs