Permutation & Combination Answers

Permutation and Combination - Quant/Math - CAT 2007

---

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 2ⁿ ways of choosing �n� objects. For e.g. if n = 3, then the three objects can be chosen in the following 2³ ways - ³C₀ ways of choosing none of the three, ³C₁ ways of choosing one out of the three, ³C₂ ways of choosing two out of the three and ³C₃ ways of choosing all three.

In the given problem, there are 5 Rock songs. We can choose them in 2⁵ 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 2⁵ - 1 = 32 - 1 = 31 ways.

Similarly, the 6 Carnatic songs, choosing at least one, can be selected in 2⁶ - 1 = 64 - 1 = 63 ways.

And the 3 Indi pop can be selected in 2³ = 8 ways. Here the option of not selecting even one Indi Pop is allowed.

Therefore, the total number of combinations = 31 * 63 * 8 = 15624