boksana wrote:
There are 5 Rock songs, 6 Pop songs, and 3 Jazz. How many different albums can be formed using the above repertoire if the albums should contain at least one Rock song and one Pop song?
A. 15,624
B. 16,384
C. 6,144
D. 384
E. 240
We can
use the answer choices to our advantage. First off, let's ignore the restriction that says "the albums should contain at least one Rock song and one Pop song"
So, we'll find the number of albums possible.
We can do so by taking the task of making albums and break it into STAGES
Let's A = 1st rock song, B = 2nd rock song, . . . . . , M = 2nd Jazz song, and N = 3rd Jazz song.
Stage 1: decide whether to include song A in album
We can choose to have song A or NOT have song A
So, we can complete this stage in
2 ways
Stage 2: decide whether to include song B in album
So, we can complete this stage in
2 ways
Stage 3: decide whether to include song C in album
So, we can complete this stage in
2 ways
.
.
.
Stage 14: decide whether to include song N in album
So, we can complete this stage in
2 ways
By the Fundamental Counting Principle (FCP), we can complete all 14 stages (and thus create an album) in
(2)(2)(2)(2)(2)(2)(2)(2)(2)(2)(2)(2)(2)(2) ways (= 16,384 ways)
NOTE: One of the 16,384 different albums includes the case in which ZERO songs are selected, which makes no sense.
So, we can subtract 1 from to get a total of 16,383 possible albums (if we IGNORE the restriction)
So, we must now subtract from 16,383 the number of albums that BREAK the restriction.
At this point,
we can use the answer choices to our advantage.
First off, we know the correct answer is LESS THAN 16,383, so we can ELIMINATE B
Now consider answer choice C (6,144). This suggests that, among the 16,383 possible albums we created, over 10,000 of them BREAK the rule that says "the albums should contain at least one Rock song and one Pop song"
Does it seem possible that well over half of the 16,383 possible albums BREAK the rule?
No, the rule doesn't seem that restrictive.
So, we can ELIMINATE answer choice C and we can eliminate D and E, since they suggest that almost all of the 16,383 possible albums BREAK the rule
We're left with A
Note: the FCP can be used to solve the MAJORITY of counting questions on the GMAT. So, be sure to learn it.
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