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# Sixteen children are trying to decide which two children will win the

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Math Expert
Joined: 02 Sep 2009
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Sixteen children are trying to decide which two children will win the  [#permalink]

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31 Jan 2017, 10:11
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Difficulty:

35% (medium)

Question Stats:

66% (01:15) correct 34% (01:15) wrong based on 253 sessions

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Sixteen children are trying to decide which two children will win the president and vice-president positions in the class. Each child can and will cast a vote for anyone in the class. If each child in the class is eligible for a position, how many different outcomes are there of the election?

A. 15
B. 30
C. 60
D. 120
E. 240

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Sixteen children are trying to decide which two children will win the  [#permalink]

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31 Jan 2017, 12:52
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Bunuel wrote:
Sixteen children are trying to decide which two children will win the president and vice-president positions in the class. Each child can and will cast a vote for anyone in the class. If each child in the class is eligible for a position, how many different outcomes are there of the election?

A. 15
B. 30
C. 60
D. 120
E. 240

Take the task of selecting a president and vice-president, and break it into stages.

Stage 1: Select someone to be president.
There are 16 children to choose from, so we can complete stage 1 in 16 ways

Stage 2: Select someone to be vice-president.
Once a child has been selected to be president, that child cannot also be vice-president.
So, there are 15 remaining children to choose from, so we can complete stage 2 in 15 ways

By the Fundamental Counting Principle (FCP), we can complete the 2 stages (and thus select a president and vice-president) in (16)(15) ways (240 ways)

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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##### General Discussion
Intern
Joined: 17 Nov 2016
Posts: 24
Re: Sixteen children are trying to decide which two children will win the  [#permalink]

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05 Jul 2018, 16:18
Quote:
By the Fundamental Counting Principle (FCP), we can complete the 2 stages (and thus select a president and vice-president) in (16)(15) ways (240 ways)

Why can't we solve it this way? 16c2=120?
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Re: Sixteen children are trying to decide which two children will win the  [#permalink]

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08 Jul 2018, 08:44
Imagine there were just 3 students - A, B, C for the post of President and Vice President.
Total no of combinations is AB, BA, AC, CA, BC, CB. i.e 3p2 ways = 6 ways.

Similarly for 16 students, it will be 16p2 = 240 ways
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Re: Sixteen children are trying to decide which two children will win the  [#permalink]

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09 Jul 2018, 19:55
Bunuel wrote:
Sixteen children are trying to decide which two children will win the president and vice-president positions in the class. Each child can and will cast a vote for anyone in the class. If each child in the class is eligible for a position, how many different outcomes are there of the election?

A. 15
B. 30
C. 60
D. 120
E. 240

We use permutations because the order is important. For example, the two outcomes (James President and Charlie Vice-President) and (Charlie President and James Vice-President) are different from each other..The number of possible outcomes is 16P2 = 16!/14! = 16 x 15 = 240.

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Re: Sixteen children are trying to decide which two children will win the  [#permalink]

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11 Jul 2018, 01:34
Bunuel wrote:
Sixteen children are trying to decide which two children will win the president and vice-president positions in the class. Each child can and will cast a vote for anyone in the class. If each child in the class is eligible for a position, how many different outcomes are there of the election?

A. 15
B. 30
C. 60
D. 120
E. 240

children no selected/not selected
1 yes
2 yes
3 no
.
.
.
16 no

Out of first two children one can be president and the other can be VP or vice-a-versa
but they are not same

total arrangement = 16!

remove arranement of 14 children not selcted
arrangement of each selected, president = 1!
VP = 1!

answer = $$\frac{16!}{14!*1!*1!}$$ = 16 * 14 = 240
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Re: Sixteen children are trying to decide which two children will win the   [#permalink] 11 Jul 2018, 01:34
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