Question

The number of ways in which $ 5\,males $ and $ 2\,\,females $ members of a committee can be seated around a round table so that the two female are not seated together is
 $
  A.\,\,480 \\
  B.\,\,600 \\
  C.\,\,720 \\
  D.\,\,840 \\
  $

Answer 1

Hint: To find required solution of the problem we first find total numbers of ways of arranging $ 7 $ persons without any restriction and then finding numbers of ways in which two females sit together and then finding their difference to find numbers of ways in which five males and two females sit around the table so that no two females sit together.

Complete step-by-step answer:
Here, it is given that $ 5\,males $ and $ 2\,\,females $ members are to be seated around a table. To find numbers of ways in which no two females are seated together around a table. We calculate the solution of a given problem in two steps.
In step one. We first calculate total arrangement of total persons irrespective or male or female around the table.
Since, we know that the number of arrangements in circular manner is always given one less than the arrangement in a straight line.
Therefore, number of ways arranging $ 7 $ persons around the table is given as:
 $
  \left( {7 - 1} \right)! \\
   \Rightarrow 6! \\
   \Rightarrow 6 \times 5 \times 4 \times 3 \times 2 \times 1 \\
   = 720 \;
  $
Therefore, number of ways of seating arrangement of $ 5\,males $ and $ 2\,\,females $ around the table without any condition is $ 720\,\,ways. $
In the second step we find a number of ways in which females always sit together.
In this we consider two female as a group of one and therefore total number of arrangements of arranging $ 6 $ around the table is given as:
\[
  \left( {6 - 1} \right)! \\
   \Rightarrow 5! \\
   \Rightarrow 5 \times 4 \times 3 \times 2 \times 1 \\
   = 120 \;
 \]
Therefore, number of ways in which two females always sit together around the table is equal to $ 120\,\,ways. $
Hence, to find the number of ways in which two females not sit together around the table will be calculated as finding the difference of first step or total ways of arranging $ 7 $ persons around the table without any restriction and second step or total ways in which two females always sit together around the table.
 $
   \Rightarrow 720 - 120 \\
   = 600 \;
  $
Hence, total numbers of ways in which $ 5\,males $ and $ 2\,\,females $ be seated around the table in such a way that no females sit together are $ 600\,\,ways. $
So, the correct answer is “Option B”.

Note: In this type of problem when arrangement is required in a circular manner. As, in a circular manner there is one way less as it can be arranged along a straight row. Which implies arrangement of n objects along a circle is given as $ \left( {n - 1} \right)! $ ways where along a row number of ways are $ n! $ . Hence, while doing such an arrangement one should carefully use the concept of arrangement to avoid any mistake.