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In how many ways can the six students be introduced?

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In how many ways can the six students be introduced?

by gmattesttaker2 » Sun Dec 22, 2013 6:38 pm
Hello,

Can you please assist with this:

A teacher has to decide in which order duets P, Q, and R will play at a school concert as
well as the order in which he will introduce each of the two students that make up a duet
before each duet plays. In how many ways can the six students be introduced?

(A) 12
(B) 24
(C) 36
(D) 48
(E) 60

OA: D


Thanks a lot,
Sri
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by GMATGuruNY » Sun Dec 22, 2013 7:23 pm
gmattesttaker2 wrote:Hello,

Can you please assist with this:

A teacher has to decide in which order duets P, Q, and R will play at a school concert as
well as the order in which he will introduce each of the two students that make up a duet
before each duet plays. In how many ways can the six students be introduced?

(A) 12
(B) 24
(C) 36
(D) 48
(E) 60

OA: D


Thanks a lot,
Sri
Let the duets be played by the following pairs:
Duet P --> A and B.
Duet Q--> C and D.
Duet R --> E and F.

There are a total of 6 positions in the concert:
PP-QQ-RR (not necessarily in this order).

Number of options for A = 6. (Any of the 6 positions in the concert.)
Number of options for B = 1. (Once A has been placed, B must be occupy the remaining position in duet P.)
Number of options for C = 4. (Any of the 4 remaining positions in the concert.)
Number of options for D = 1. (Once C has been placed, D must occupy the remaining position in duet Q.)
Number of options for E = 2. (Either of the 2 remaining positions in the concert.).
Number of options for F = 1. (Only one position left.)
To combine these options, we multiply:
6*1*4*1*2*1 = 48.

The correct answer is D.
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by Brent@GMATPrepNow » Mon Dec 23, 2013 7:57 am
gmattesttaker2 wrote:Hello,

Can you please assist with this:

A teacher has to decide in which order duets P, Q, and R will play at a school concert as
well as the order in which he will introduce each of the two students that make up a duet
before each duet plays. In how many ways can the six students be introduced?

(A) 12
(B) 24
(C) 36
(D) 48
(E) 60

Here's another approach . . .

Take the task of introducing the performers and break it into stages.

Stage 1: Determine the order of the duets (disregarding, for the moment, the order of the individual singers)
There are 3 duets, so we can arrange them in 3! ways ( = 6 ways)

Stage 2: Determine the order of the two singers in the first duet to perform
We can complete this stage in 2 ways.

Stage 3: Determine the order of the two singers in the second duet to perform
We can complete this stage in 2 ways.

Stage 4: Determine the order of the two singers in the third duet to perform
We can complete this stage in 2 ways.

By the Fundamental Counting Principle (FCP), we can complete all 4 stages (and thus arrange all 6 performers) in (6)(2)(2)(2) ways ([spoiler]= 48 ways = D[/spoiler])

Cheers,
Brent

Aside: For more information about the FCP, watch our free video: https://www.gmatprepnow.com/module/gmat-counting?id=775
Brent Hanneson - Creator of GMATPrepNow.com
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