Let x and y be distinct integers where 1 ≤ x ≤ 25 and 1 ≤ y ≤ 25 . Then, the number of ways of choosing x and y , such that x + y is divisible by 5 , is _____ .

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Given: 1 ≤ x , y ≤ 25 And, x , y are distinct integers. Let x + y = 5 k , where k ∈ N So, we have x y Number of ways 5 λ i.e, 5 , 10 , 15 , 20 , 25 5 λ i.e, 5 , 10 , 15 , 20 , 25 Since, x and y are distinct integers, so we cannot pair 5 , 5 , 10 , 10 , . . . 25 , 25 20 5 λ + 1 i.e., 1 , 6 , 11 , 16 , 21 5 λ + 4 i.e., 4 , 9 , 14 , 19 , 14 25 5 λ + 2 i.e, 2 , 7 , 12 , 17 , 22 5 λ + 3 i.e., 3 , 8 , 13 , 18 , 23 25 5 λ + 3 i.e., 3 , 8 , 13 , 18 , 23 5 λ + 2 i.e, 2 , 7 , 12 , 17 , 22 25 5 λ + 4 i.e., 4 , 9 , 14 , 19 , 14 5 λ + 1 i.e, 1 , 6 , 11 , 16 , 21 25 Total number of ways = 120

$x$	$y$		Number of ways
$5 λ$ i.e, $5, 10, 15, 20, 25$	$5 λ$ i.e, $5, 10, 15, 20, 25$	Since, $x$ and $y$ are distinct integers, so we cannot pair $(5, 5), (10, 10), . . . (25, 25)$	$20$
$5 λ + 1$ i.e., $1, 6, 11, 16, 21$	$5 λ + 4$ i.e., $4, 9, 14, 19, 14$		$25$
$5 λ + 2$ i.e, $2, 7, 12, 17, 22$	$5 λ + 3$ i.e., $3, 8, 13, 18, 23$		$25$
$5 λ + 3$ i.e., $3, 8, 13, 18, 23$	$5 λ + 2$ i.e, $2, 7, 12, 17, 22$		$25$
$5 λ + 4$ i.e., $4, 9, 14, 19, 14$	$5 λ + 1$ i.e, $1, 6, 11, 16, 21$		$25$

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