Question

There are 10 girls and 8 boys in a class room including Mr. Ravi, Ms. Rani and Ms. Radha. a list of speakers consisting of $8$ girls and $6$ boys has to be prepared. Mr. Ravi refuses to speak if Ms. Rani is the speaker. Ms. Rani refuses to speak if Ms. Radha is a speaker.
The number of ways the list can be prepared is?
$A)202$
$B)308$
$C)567$
$D)952$

Answer 1

Hint: Since the question is to find the number of ways, we are going to use permutation and combination methods which we will study on our schools to approach the given questions to find the number of ways since the number of permutations of r-objects can be found from among n-things is ${}^n{p_r}$ where p refers to the permutation. Also, similarly, for combination we have r-objects and among n-total things are ${}^n{C_r}$.
Formula used: ${}^n{C_r} = \dfrac{{n!}}{{r!(n - r)!}}$

Complete step-by-step solution:
Let the question is to find the different number of ways hence we use only a combination method to approach the given question.
Case1: When Ms. Radha is the speaker.
Ms. Rani would not speak. In this case, the total number of selected girls are only seven out of eight girls (Radha is the speaker and Rani will not speak, so Radha will be one speaker and we need to select the balance of seven from the remaining eight unknown girls)
But for boys it has not changed and remains at six out of eight.
Thus, we get, \[{}^8{C_7} \times {}^8{C_6} = \dfrac{{8!}}{{7!1!}} \times \dfrac{{8!}}{{6!2!}}\]
Further solving this we get, \[{}^8{C_7} \times {}^8{C_6} = 8 \times 28 \Rightarrow 224\]
Case2: When Ms. Radha is not the speaker, but Ms. Rani is the speaker, Ravi would not be speaking
In this case, we have the select the balance girls in seven out of eight (since Rani is the speaker and Radha is not, so two is done)
For boys, we have six out of seven (since Ravi is not the speaker)
Thus, we get, \[{}^8{C_7} \times {}^7{C_6} = \dfrac{{8!}}{{7!1!}} \times \dfrac{{7!}}{{6!1!}}\]
Further solving we get, \[{}^8{C_7} \times {}^7{C_6} = 8 \times 7 \Rightarrow 56\]
Case3: In this case, Neither Ms. Radha nor Rani is the speaker (if Radha speaks then Rani will not if Rani speaks then Radha will not)
There are eight girls out of eight girls. And for boys, there will be no change.
Thus, we get, \[{}^8{C_8} \times {}^8{C_6} = \dfrac{{8!}}{{8!0!}} \times \dfrac{{8!}}{{6!2!}}\]
Further soling we get, \[{}^8{C_8} \times {}^8{C_6} = 28\]
Hence in total we get $224 + 56 + 28 = 308$ways.
Hence option $B)308$ is correct.

Note: If there is no restriction like Mr. Ravi refuses to speak if Ms. Ravi is the speaker. Ms. Rani refuses to speak if Ms. Radha is a speaker. We are simply able to apply the formula using the information that there are $10$girls and $8$ girls that need to speak and from $8$boys, $6$boys have to speak.
Thus, we get from the combination formula for the total number of ways is \[{}^{10}{C_8} \times {}^8{C_6}\]
Which can be repressed as \[{}^{10}{C_8} \times {}^8{C_6} = \dfrac{{10!}}{{8!2!}}\dfrac{{8!}}{{6!2!}} = 1260\](after the simplification)
Thus, we get $1260$ways if there is no restriction.