Question

P, Q, R, and S have to give lectures to an audience. The organizer can arrange the order of their presentation in?
A. 4 ways
B. 12 ways
C. 256 ways
D. 24 ways

Answer 1

Hint: In our case, we are provided that \[P,Q,R\] and \[S\] are delivering lectures to the audience and we are to determine the number of ways the organizer can arrange their lecture presentation. To determine this, we have to use the permutation formula. As it is given that there are four people, then we have to write the combination according to the given data that \[n = 4!\] to determine the desired answer.

Formula Based:
The formula used is the permutation formula:
\[n! = n \times \left( {n - 1} \right) \times \left( {n - 2} \right) \times ...{\rm{ }} \times 1\]
where n is the number of items to be arranged.

Complete step by step solution:
Considering that, \[P,Q,R\] and \[S\] have to give lectures to an audience
And we have to calculate the number of ways the organizer can arrange the order of the presentation.
Now, let us arrange the order of the lectures, using permutation.
As there are four people, the permutation will be
\[n = 4!\]
Now, let us arrange the order of the lectures, using permutation.
As there are four people, the permutation will be n!
where n is the number of items in the set to be arranged.
In this case, \[n = 4\]
\[n! = n \times \left( {n - 1} \right) \times \left( {n - 2} \right) \times ...{\rm{ }} \times 1\]
\[n! = 4 \times \left( {4 - 1} \right) \times \left( {4 - 2} \right) \times \left( {4 - 3} \right)\]
\[ = 4{\rm{ }} \times 3 \times 2 \times 1\]
Now, on multiplying the above expression, we get
\[ = 24\]
So, there are \[24\] different ways to arrange the four people \[P,Q,R,S\] to conduct lectures.
Therefore, the organizer can arrange the order of their presentation in \[24\] ways

Hence, the option D is correct

Note: Students should keep in mind that a permutation is an orderly arrangement of outcomes as well as an arranged combination. So, using the permutation formula, we have determined the total number of various ways to arrange n different objects. So, one should be very careful in finding the permutation. Students should also remember that the factorial of numbers is another key topic to understand in permutations. The first n positive numbers are said to a factorial.