JEE Main · 2023 · 25 Jan 2023 · Shift 1 · Permutations and Combinations

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 ______.

Answer: 120

Solution

Splitting

1

25

by remainder on division by

5

gives five classes of five numbers each. For

x + y

to be a multiple of

5

with

x \neq = y

, the ordered pairs come from residues

(0, 0)

(1, 4)

(4, 1)

(2, 3)

and

(3, 2)

. The

(0, 0)

class contributes

5 \times 4 = 20

ordered pairs, and each of the other four contributes

5 \times 5 = 25

, giving

20 + 100 = 120

JEE Main · 2023 · 25 Jan 2023 · Shift 1 · Integer answer