In a cycle race there are 5 persons named as J, K, L, M, N participated for 5…

Key insight: exactly half of all possible orderings have M before N.

Method 1 (symmetry): There are 5! = 120 total orderings. By symmetry, M is before N in half of them, so 120 / 2 = 60.

Method 2 (position counting): Fix M's position and count choices for N (N must come after M). For each choice of positions for M and N, the remaining three riders can be arranged in 3! ways:

• If M is 1st: N can be in any of 4 later positions → 4 × 3! = 24.

• If M is 2nd: N has 3 choices after M → 3 × 3! = 18.

• If M is 3rd: N has 2 choices → 2 × 3! = 12.

• If M is 4th: N has 1 choice → 1 × 3! = 6.

• If M is 5th: N cannot be after M → 0.

Total = 24 + 18 + 12 + 6 = 60.