A hall has 14 gates. In how many ways, can a man enter the hall through one…

Concept

When a task is carried out in stages performed one after another, the total number of ways is the product of the number of choices available at each stage — the fundamental counting principle. If both stages draw from the same pool of k distinct objects and the second stage may not repeat the object chosen at the first, then the first stage offers k choices and the second offers only k − 1, so the number of ordered pairs of distinct objects is k × (k − 1).

This count is exactly the number of ordered selections of 2 distinct objects out of k. Order is part of the arrangement, so a pair taken in one sequence is counted separately from the same pair taken in the opposite sequence.

Applying it here

1. The pool here is the hall's 14 gates, and the man performs two ordered stages: he picks the gate he enters by, and then the gate he leaves by.

2. Entry stage: any gate of the hall may be used, so there are 14 choices.

3. Exit stage: the gate already used to enter is barred, so 13 of the 14 gates remain and there are 13 choices.

4. By the counting principle the two stages multiply: 14 × 13 = 182 ways.

Cross-check

Count it the other way round as a check. Ignoring the restriction there are 14 × 14 = 196 ordered (entry gate, exit gate) pairs. Exactly 14 of these use the same gate twice, one for each gate, so the admissible count is 196 − 14 = 182, which matches the direct count.

In permutation notation the same figure is 14P2 = 14! / 12! = 14 × 13 = 182.