CMI2017-A-02

Question

An FM radio channel has a repository of $10$ songs. Each day, the channel plays $3$ distinct songs that are chosen randomly from the repository.

Mary decides to tune in to the radio channel on the weekend after her exams. What is the probability that no song gets repeated during these $2$ days?

• A. $\begin{pmatrix} 10\\ 3 \end{pmatrix}^{2}*\begin{pmatrix} 10\\ 6 \end{pmatrix}^{-1}$
• B. $\begin{pmatrix} 10\\ 6 \end{pmatrix}*\begin{pmatrix} 10\\ 3 \end{pmatrix}^{-2}$
• C. $\begin{pmatrix} 10\\ 3 \end{pmatrix}*\begin{pmatrix} 7\\ 3 \end{pmatrix}*\begin{pmatrix} 10\\ 3 \end{pmatrix}^{-2}$
• D. $\begin{pmatrix} 10\\ 3 \end{pmatrix}*\begin{pmatrix} 7\\ 3 \end{pmatrix}*\begin{pmatrix} 10\\ 6 \end{pmatrix}^{-1}$

Comments

Here Order is important

ie.  (1,2,3 ) (4,5,6)  != (4,5,6 ) ( 1,2,3)

option B doesn't consider this order that's why B is eliminated.

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$^{10}C_3$ is the number of ways of choosing 3 songs on day 1. $^7C_3$ is the number of ways of choosing 3 different songs on day 2, so $^{10}C_3$.$^7C_3$ is the number of combinations that meet Mary’s requirement. $^{10}C_3$ $^{10}C_3$ is the total number of ways of choosing 3 songs on each of the two days, without any constraints.

$P=\dfrac{^{10}C_3}{^{10}C_3}\times\dfrac{^7C_3}{^{10}C_3}$

Answer 1

Answer is (C)

Mary listen to song on weekend $->$ $2$ days

Selecting $3$ songs from $10$ can be done by $10C3$ ways

Now we select $3$ songs from remaining $7$ song in $7C3$

If we select this way we are sure that Mary will listen new song only

Probability can be given by =$\frac{^{10}C_{3}*^{7}C_{3}}{^{10}C_{3}*^{10}C_{3}}$

Comments

Why is (D) not the correct answer?

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@chauhansunil20th it will imply that we are selecting 6 songs out of 10 which is incorrect

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10C3 does not mean that we allow repeatations means same song can be played 3 times?

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(D) will not be correct because it will imply that we are choosing 6 distinct songs on two days. So, by choosing this way, we indirectly imply that the songs that we chose on day 1 are completely different than the songs we chose on day 2, which is incorrect. Hence, D is not correct. 

Answer 2

Comments

nice explanation