Permutation and Combination  Quant/Math  CAT 2016
Question 4 the day:
June 10, 2002
The question for the day is from the topic Permutation and Combination.
Question:
 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?
(1) 15624
(2) 16384
(3) 6144
(4) 240
Correct Answer  (1)
Solution:
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  ^{3}C_{0} ways of choosing none of the three, ^{3}C_{1} ways of choosing one out of the three, ^{3}C_{2} ways of choosing two out of the three and ^{3}C_{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 = 15624
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