Let x and y be distinct integers where 1 ≤ x ≤ 25 and 1 ≤ y ≤ 25 . Then, the number…

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 _____ .

Solution

Given:

1≤x,y≤25

And, x,y are distinct integers.

Let

x+y=5k, where k∈N

So, we have

x y   Number of ways
5λ i.e, 5, 10, 15, 20, 25 5λ 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
 5λ+1 i.e., 1, 6, 11, 16, 21  5λ+4 i.e., 4, 9, 14, 19, 14   25
 5λ+2 i.e, 2,7, 12, 17, 22  5λ+3 i.e., 3, 8, 13, 18, 23   25
 5λ+3 i.e., 3, 8, 13, 18, 23 5λ+2 i.e, 2, 7, 12, 17, 22   25
 5λ+4 i.e., 4, 9, 14, 19, 14 5λ+1 i.e, 1, 6, 11, 16, 21   25

Total number of ways =120

Asked in: JEE Main 2023 (25 Jan Shift 1)

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