Question

Four boys and three girls stand in a queue for an interview. Find the probability that they will be in an alternate position.
A.\[\dfrac{1}{{34}}\]
B.\[\dfrac{1}{{35}}\]
C.\[\dfrac{1}{{17}}\]
D.\[\dfrac{1}{{68}}\]

Answer 1

Hint: In order to solve the question, first find the number of ways boys and girls can be arranged. Next, we will find the total possible arrangements. Then we will find the probability that they will be in an alternate position.
Formula used :
\[{\rm{Probability of an event}} = \dfrac{{{\rm{favorable outcome}}}}{{{\rm{total number of outcome}}}}\]
Complete step by step solution:
Given that
The total number of boys is 4.
The total number of girls is 3.
Here it is given that they need to be in an alternate position, that is
\[{\rm{B G B G B G B}}\]
So here the boys can be arranged in \[4!\] ways and girls can be arranged in \[3!\] ways.
Hence the total possible arrangements
\[ = 4! + 3!\]
\[ = 7!\]
Since we know that
\[{\rm{Probability of an event}} = \dfrac{{{\rm{favorable outcome}}}}{{{\rm{total number of outcome}}}}\]
Hence the required probability
\[ = \dfrac{{4!{\rm{ }} \times {\rm{ }}3!}}{{7!}}\]
\[ = \dfrac{{4 \times 3 \times 2 \times 1 \times 3 \times 2 \times 1}}{{7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}}\]
\[ = \dfrac{1}{{35}}\]
Therefore the probability that they will be in an alternate position is \[\dfrac{1}{{35}}\].

Hence option B is the correct answer.
Note: Students can make mistakes while interpreting the meaning of the question. Always remember that by simply dividing the favourable number of possibilities by the entire number of possible outcomes, the probability of an occurrence can be determined using the probability formula. Also, here the order of arrangement of the students is important, so we use the permutation.