Moderate Permutation-Combination Solved QuestionAptitude Discussion

 Q. There are 5 Rock songs, 6 Carnatic songs and 3 Indi pop songs. How many different albums can be formed using the above repertoire if the albums should contain at least 1 Rock song and 1 Carnatic song?
 ✔ A. $15,624$ ✖ B. $16,384$ ✖ C. $6,144$ ✖ D. $240$

Solution:
Option(A) is correct

There are $2^n$ ways of choosing $n$ objects.

For e.g. if $n = 3$, then the three objects can be chosen in the following $2^3$ ways:

${^3C_0}$ ways of choosing none of the three,

${^3C_1}$ ways of choosing one out of the three,

${^3C_2}$ ways of choosing two out of the three and

${^3C_3}$ ways of choosing all three.

In the given problem, there are 5 Rock songs. We can choose them in $2^5$ ways. However, as the problem states that the case where you do not choose a Rock song does not exist (at least one rock song has to be selected), it can be done in:

$2^{5} - 1$

$= 32 - 1$

$= 31$ ways.

Similarly, the 6 Carnatic songs, choosing at least one, can be selected in:

$2^6 - 1$

$= 64 - 1$

$= 63$ ways.

And the 3 Indi pop can be selected in:

$2^3 = 8$ ways.

Here the option of not selecting even one Indi Pop is allowed.

Therefore, the total number of combinations,

$= 31 × 63 × 8$

$= \textbf{15624}$