Question

There are 8 candidates contesting for a certain constituency in an election.
In how many ways their names can be listed on a ballot paper?
(a) 720
(b) 5040
(c) 1680
(d) None

Answer 1

This is a classical problem of arrangement. We are said that there are 8 candidates and there is a list in which names of all the 8 are to be listed. This list is a linear list and there are a well-defined start and end of the list. All 8 people are different and can take anyplace. Therefore, we will occupy the places one by one and find the total number of ways in which the names of these candidates can occupy places in the list.

Complete step-by-step solution:
The number of candidates contesting for a certain constituency in an election is 8. This list is a linear list and there are a well-defined start and end of the list. All 8 people are different and can take anyplace.
Now the first place can be occupied in 8 ways as there are 8 members and no name has been entered in the list as of yet.
$\Rightarrow$ 8 _ _ _ _ _ _ _
Now, one of the 8 names will occupy the first place. Now, the remaining names of the candidates are 7. Thus, the second place can be occupied in 7 ways.
$\Rightarrow$ 8 7 _ _ _ _ _ _
After two places are occupied, 6 names are left to be listed.
The third-place thus can be filled in 6 ways.
$\Rightarrow$ 8 7 6 _ _ _ _ _
Now 5 candidates are left and so, 4th place can be filled in 5 ways.
$\Rightarrow$ 8 7 6 5 _ _ _ _
Therefore, in a similar fashion, we can say that the 5th place will be filled in 4 ways, 6th place in 3 ways, 7th place can be filled in 2 ways and for the 8th place, there will be only one name and hence, only in one way.
$\Rightarrow$ 8 7 6 5 4 3 2 1
Therefore, the number of ways 8 names can be arranged in 8 places will be given as $8\times 7\times 6\times 5\times 4\times 3\times 2\times 1=40320$
Hence, option (d) is the correct option.

Note: Students can directly remember that the number of ways to arrange n items in n places is given by n! where n! = n(n – 1)(n – 2)(n – 3)…1. If there are n items to be arranged in r places, then the number of ways they can be arranged is given by $^{n}{{P}_{r}}$, where $^{n}{{P}_{r}}=\dfrac{n!}{\left( n-r \right)!}$. There are valid only when the things are to be arranged in a linear list with a well-defined start and end.