permutation/combination

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vaibhav101: Master | Next Rank: 500 Posts; Posts: 138; Joined: Mon May 01, 2017 11:56 pm; Thanked: 4 times

permutation/combination

by vaibhav101 » Sat Sep 01, 2018 6:05 am

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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 album should contain at least 1 rock song and 1 carnatic song?

A 15624
B 16384
C 6144
D 240
E 8123

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Source: Beat The GMAT — Problem Solving | Sat Sep 01, 2018 6:05 am

deloitte247: Legendary Member; Posts: 2214; Joined: Fri Mar 02, 2018 2:22 pm; Followed by:5 members

by deloitte247 » Sat Sep 08, 2018 10:33 am

There are 5 rock songs,
6 Carnatic songs, and
3 Indi pop songs
Each of these songs can either be included in the album or not included in the album, so we have the total of 2 options for each song.
For 5 rock songs in which the album must certain at least 1 rock song.
Total number of different rock songs combination in the album,
$$=\left(2^5\right)-1$$
=32 - 1 = 31 combinations,
Where at least one rock song is present in the albums.
For 6 Carnatic songs in which the album must curtain at least 1 Carnatic song;
Total number of different Carnatic songs combination in the album
$$=\left(2^6\right)-1$$
=64 - 1
= 63 combinations where at least one Carnatic song is present in the albums.
For 3 Indi pop songs in which each of them can either be included in the albums or not included,
Total number of different Indi pop songs combination in the album,
$$=2^3$$
= 8 combinations which includes one combination where none of the Indi pop songs is included in the album.
Total Albums = 31 * 63 * 8
= 15, 624

Option A is CORRECT.

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pm; Location: Vancouver, BC; Thanked: 5254 times; Followed by:1268 members; GMAT Score:770

by Brent@GMATPrepNow » Sun Sep 09, 2018 9:27 am

vaibhav101 wrote: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 album should contain at least 1 rock song and 1 carnatic song?

A 15624
B 16384
C 6144
D 240
E 8123

This question is a little too ambiguous to be an official GMAT question.
Let's say X, Y and Z represents 3 different songs.
Is an album with song X first, song Y second and song Z third considered DIFFERENT FROM an album with song Z first, song X second and song Y third?
The answer to this question is not apparent in the wording.

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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Scott@TargetTestPrep: GMAT Instructor; Posts: 8086; Joined: Sat Apr 25, 2015 10:56 am; Location: Los Angeles, CA; Thanked: 43 times; Followed by:29 members

by Scott@TargetTestPrep » Mon Sep 17, 2018 5:31 pm

vaibhav101 wrote: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 album should contain at least 1 rock song and 1 carnatic song?

A 15624
B 16384
C 6144
D 240
E 8123

Each song can either be included in the album or excluded from the album, so each song has two choices. Since there are 14 songs, the number ways to form an album is 2^14 = 16,384 if there are no restrictions. (Notice that 2^14 is also equal to 2^5 x 2^6 x 2^3 where the exponents 5, 6 and 3 are the number of rock, carnatic, and indi pop songs, respectively.)

However, the album must contain at least 1 rock song and 1 carnatic song. Of the 2^5 = 32 ways to include any number (from 0 to 5) of rock songs, there is only 1 way that no rock song is included, so the number of ways to include at least 1 rock song is 32 - 1 = 31. Similarly, Of the 2^5 = 64 ways to include any number (from 0 to 6) of carnatic songs, there is only 1 way that no carnatic song is included, so the number of ways to include at least 1 carnatic song is 64 - 1 = 63. Note that there are no restrictions on the number of indi pop songs, so we have 2^3 = 8 ways to choose the indi pop songs. Therefore, the number of ways to include at least 1 rock song and 1 carnatic song in the album is:

31 x 63 x 8 = 15,624

Answer: A

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