Question

There are \[8\] chairs in a room on which \[6\] persons are to be seated; out of which one is a guest with one specific chair. In how many ways can they sit?
A.\[2520\]
B.\[60480\]
C.\[30\]
D.\[346\]

Answer 1

Hint: Permutation relates to the act of arranging all the members of a set into some sequence or order. The combination is a way of selecting items from a collection, such that (unlike permutations) the order of selection does not matter. In smaller cases, it is possible to count the number of combinations.

Complete step-by-step answer:
A permutation is the choice of \[r\] things from a set of \[n\] things without replacement and where the order matters.
\[{}^n{P_r} = \dfrac{{n!}}{{(n - r)!}}\]
A combination is the choice of \[r\] things from a set of \[n\] things without replacement and where order doesn't matter.
\[{}^n{C_r} = \dfrac{{n!}}{{r!(n - r)!}}\]
There are \[8\] chairs and \[6\] persons are to be seated.
One is a guest with one specific chair.
Thus, we have \[7\] chairs and \[5\] persons remaining.
Since we have \[7\]chairs and \[5\] persons are to be seated on them. Therefore we have \[^7{C_5}\] choices.
Since the \[5\] persons can sit in any order. Therefore we have \[5!\] choices.
Therefore required number of ways \[ = {}^7{C_5} \times 5! = 21\times 120 = 2520\]
So, the correct answer is “Option A”.

Note: The combination is a way of selecting items from a collection, such that (unlike permutations) the order of selection does not matter. A permutation is used for the list of data (where the order of the data matters) and the combination is used for a group of data (where the order of data doesn’t matter). One must know the formula for permutation and comn\binations and should have accuracy in calculations.