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Question: Given \(4\) gentlemen and \(4\) ladies take seats at random round a table. The probability they are ...

Given 44 gentlemen and 44 ladies take seats at random round a table. The probability they are sitting alternately is
A.435\dfrac{4}{35}
B.170\dfrac{1}{70}
C.235\dfrac{2}{35}
D.135\dfrac{1}{35}

Explanation

Solution

We are given that, 44 gentlemen and 44 ladies take seats at random round a table. Now let us assume that the gentlemen take their seat first. So, the number of ways gentlemen will take their seats will be=(4−1)!=3!=(4-1)!=3!.Now in the remaining gaps that are between two gentlemen the lady will be seated.
The number of ways the lady seat in gaps is=4!=4!. After that, find the total number of ways seating alternately. Find the total probability that 44 gentlemen and 44 ladies take seats at random round a table.

Complete step-by-step answer:
We are given that, 44 gentlemen and 44 ladies take seats at random round a table.
Now let us assume that the gentlemen take their seat first.
So, the number of ways gentlemen will take their seat will be=(4−1)!=3!=(4-1)!=3!.
Now in remaining gaps that is between two gentlemen the lady will be seating.
The number of ways the lady seat in gaps is=4!=4!
Total number of ways seating alternately=(3!)×(4!)=(3!)\times (4!)
Now, the number of total ways the44 gentlemen and 44 ladies take seats at random round a table is=(8−1)!=7!=(8-1)!=7!
Hence, the probability they are sitting alternately is=3!×4!7!=\dfrac{3!\times 4!}{7!}
Now simplifying above we get,
The probability they are sitting alternately is=3!×4!7×6×5×4!=\dfrac{3!\times 4!}{7\times 6\times 5\times 4!}
Again, simplifying we get,
The probability they are sitting alternately is=17×5=\dfrac{1}{7\times 5}
The probability they are sitting alternately is=135=\dfrac{1}{35}
Therefore, 44 gentlemen and 44 ladies take seats at random round a table. The probability they are sitting alternately is 135\dfrac{1}{35}.
The correct answer is option (D).

Note: Probability means possibility. It is a branch of mathematics that deals with the occurrence of a random event. The value is expressed from zero to one. Probability has been introduced in Maths to predict how likely events are to happen. The probability formula is defined as the possibility of an event to happen is equal to the ratio of the number of favourable outcomes and the total number of outcomes.