Question

if there are 5 persons who want to get off a lift in any one of the 7 floors such that each goes to a different floor then find number of ways

Answer 1

2520

Answer 2

The problem asks us to find the number of ways 5 persons can get off a lift in any one of 7 floors such that each person goes to a different floor. This is a permutation problem because the order in which persons choose floors matters, and each person must choose a unique floor.

We have 7 distinct floors available and 5 distinct persons.
Let's consider the choices for each person:

1. The first person has 7 options for the floor to get off.
2. Since the second person must get off at a different floor than the first, there are 6 remaining options for the second person.
3. For the third person, there are 5 remaining floor options (different from the first two).
4. For the fourth person, there are 4 remaining floor options.
5. For the fifth person, there are 3 remaining floor options.

By the fundamental principle of counting (product rule), the total number of ways is the product of the number of choices for each person:
Number of ways = $7 \times 6 \times 5 \times 4 \times 3$

Calculating the product:
$7 \times 6 = 42$
$42 \times 5 = 210$
$210 \times 4 = 840$
$840 \times 3 = 2520$

Alternatively, this can be solved using the permutation formula $P(n, k) = \frac{n!}{(n-k)!}$, where 'n' is the total number of items to choose from (7 floors) and 'k' is the number of items to choose (5 persons).
$P(7, 5) = \frac{7!}{(7-5)!} = \frac{7!}{2!} = \frac{7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{2 \times 1} = 7 \times 6 \times 5 \times 4 \times 3 = 2520$.

The total number of ways is 2520.