Question

What is the number of ways that 4 boys and 3 girls can be seated so that boys and girls alternate?
A.12
B.72
C.120
D.144

Answer 1

Hint: To answer this topic, we'll look at some key concepts, such as the fundamental principle of multiplication and permutations and combinations. After that, we will make 4 boys to be seated alternatively in the row and arrange them in \[4!\] ways. Then 3 Girls can be seated alternatively in the row and arrange them in \[3!\] ways by using the formula \[n!=n\times (n-1)\times (n-2)!\]. Then we'll multiply both results to find the total number of ways the boy and girl can be seated in a different order.

Complete answer:Given:
We need to figure out how to seat four girls and three boys in a row so that no two boys are in the same row.
Before we go any further, it's crucial to understand the following key concepts and formulas:
Fundamental Principle of Multiplication: If there are two jobs such that one of them can be completed in m ways, and when it has been completed in any of these \[m\] ways the second job can be completed in \[n\] ways then two jobs in succession can be completed in \[m\times n\]ways.
Number of linear arrangements of \[r\] distinct objects will be equal to \[r!\]
Above concept and fundamental principle will be used in this problem.
Now, we will make 4 boys and 3 girls to be seated alternatively in the row. For more clarity, look at the figure given below:

\[{{B}_{1}}\]

\[{{G}_{1}}\]

\[{{B}_{2}}\]

\[{{G}_{2}}\]

\[{{B}_{3}}\]

\[{{G}_{3}}\]

\[{{B}_{4}}\]

In the above figure\[{{B}_{1}},{{B}_{2}},{{B}_{3}},{{B}_{4}}\] are the 4 boys and \[{{G}_{1}},{{G}_{2}},{{G}_{3}}\] are the girls we can arrange in the 4 boys and 3 girls can be alternatively.
Now 4 boys can be arranged among themselves be \[m=4!=24\] ways
And 3 girls can also arrange among themselves in \[n=3!=6\] ways
Now, from the fundamental principle of multiplication, we can say that the number of ways in which 4 boys and 3 girls are seated in a row so that boys and girls are seated alternatively will be equal to \[m\times n=24\times 6=144\] ways.
Thus, the required number of ways will be 144 ways.
So, the correct option is “option D”.

Note:
Here, the learner must first comprehend what is being asked in the question before proceeding with the proper strategy to swiftly obtain the correct answer. And in such permutation and combination questions. Apply the fundamental concepts and use figures to illustrate the event.
Moreover, for objective type problems, we could have used the formula \[\dfrac{g!(g+1)!}{(g+1-b)!}\]directly where \[g\] is the number of girls and b is the number of ways.