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# In how many ways can 3 people out of 14 people be given 3 prizes(gold,

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In how many ways can 3 people out of 14 people be given 3 prizes(gold,  [#permalink]

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Updated on: 17 Mar 2019, 03:18
6
00:00

Difficulty:

45% (medium)

Question Stats:

59% (01:39) correct 41% (02:02) wrong based on 37 sessions

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In how many ways can 3 people out of 14 people be given 3 prizes(gold,silver,bronze) so that Jack is awarded? Jack is one of 14 people.

A) 2184
B) 546
C) 468
D) 364
E) 156

but I have not been able to understand how.

The direction that I proceeded was:
Jack gets gold or jack gets silver or jack gets bronze
=$$13*12 + 13*12 + 13*12$$

This seems to be wrong,so if somebody could explain where I'm going wrong and how to arrive at the correct answer it would be great!

_________________

Originally posted by VandyonWheels on 07 Nov 2012, 06:53.
Last edited by Bunuel on 17 Mar 2019, 03:18, edited 2 times in total.
Renamed the topic.
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Re: In how many ways can 3 people out of 14 people be given 3 prizes(gold,  [#permalink]

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07 Nov 2012, 07:03
In how many ways can 3 people out of 14 people be given 3 prizes(gold,silver,bronze) so that Jack is awarded? Jack is one of 14 people.

The number of groups of 3 to receive prizes is $$C^2_{13}=78$$.

Each of these groups of 3 can be arranged in 3!=6 ways, thus the total ways is 78*6=468.

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Re: In how many ways can 3 people out of 14 people be given 3 prizes(gold,  [#permalink]

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07 Nov 2012, 07:10
Brunel,

I'm having difficulty understanding why you multiply by 3! Instead
Of simply 3.

Thank you.

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Re: In how many ways can 3 people out of 14 people be given 3 prizes(gold,  [#permalink]

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07 Nov 2012, 07:20
1
Actually, I arrived at
The answer a different way. I used
The permutation formula 13!\11! To get 156, and then multiples by 3 for the 3 different ways jack could place.

Posted from my mobile device
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Re: In how many ways can 3 people out of 14 people be given 3 prizes(gold,  [#permalink]

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07 Nov 2012, 07:22
aceon wrote:
Brunel,

I'm having difficulty understanding why you multiply by 3! Instead
Of simply 3.

Thank you.

Posted from my mobile device

Say one particular group is Jack, X and Y. Now, JXY can be arranged in 3! ways to receive gold, silver, or bronze:

GSB:
JXY
JYX
XJY
XYJ
YJX
YXJ

Hope it's clear.
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Re: In how many ways can 3 people out of 14 people be given 3 prizes(gold,  [#permalink]

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27 Dec 2012, 22:02
1
My approach:

Since I know that Jack is awarded and another 2 so I have: J _ _

How many ways to select 2 from the remaining 13? 13!/2!1! = 78 Thus, this means there are 78 possible selection of winners from the 14. All of which includes Jack and some other 2 persons.

How many ways to award a group? We can imagine ranking three people as first, second and third. Thus we can get the possible arrangements of 3 people. 3! = 6

78*6 = 468
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Re: In how many ways can 3 people out of 14 people be given 3 prizes(gold,  [#permalink]

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07 Aug 2016, 21:39
Hey! I tried an alternate approach where the no. Of ways of choosing prize winners is 1*13*12 and no of permutations is 3! So my answer is 3!*1*13*12
Where am I going wrong here though

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Re: In how many ways can 3 people out of 14 people be given 3 prizes(gold,  [#permalink]

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22 Mar 2017, 20:48
VandyonWheels wrote:
In how many ways can 3 people out of 14 people be given 3 prizes(gold,silver,bronze) so that Jack is awarded? Jack is one of 14 people.

468

but I have not been able to understand how.

The direction that I proceeded was:
Jack gets gold or jack gets silver or jack gets bronze
=$$13*12 + 13*12 + 13*12$$

This seems to be wrong,so if somebody could explain where I'm going wrong and how to arrive at the correct answer it would be great!

Lets ensure Jack gets a prize - 1C1
Choose any 2 from the remaining 13 get prizes - 13C2

Ways of distributing 3 prizes among 3 people - 3!

So total number of ways - $$1C1 * 13C2 * 3!$$ = $$1*13*6*6$$ = $$468$$
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Re: In how many ways can 3 people out of 14 people be given 3 prizes(gold,  [#permalink]

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17 Mar 2019, 00:16
2
VandyonWheels wrote:
In how many ways can 3 people out of 14 people be given 3 prizes(gold,silver,bronze) so that Jack is awarded? Jack is one of 14 people.

468

but I have not been able to understand how.

The direction that I proceeded was:
Jack gets gold or jack gets silver or jack gets bronze
=$$13*12 + 13*12 + 13*12$$

This seems to be wrong,so if somebody could explain where I'm going wrong and how to arrive at the correct answer it would be great!

Since Jack has to be given an award so we need two more individuals out of rest 13 candidates

Those two candidates may be selected in 13C2 = 78 ways

Now we have team of 3 candidates (Jack included) and three persons can be given three different rewards in 3*2*1 = 3! = 6 ways

Total favourable ways = 78*6 = 468

Bunuel The following options may be given to the question (with C being the correct Answer)

A) 2184
B) 546
C) 468
D) 364
E) 156
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Re: In how many ways can 3 people out of 14 people be given 3 prizes(gold,  [#permalink]

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17 Mar 2019, 03:19
GMATinsight wrote:
VandyonWheels wrote:
In how many ways can 3 people out of 14 people be given 3 prizes(gold,silver,bronze) so that Jack is awarded? Jack is one of 14 people.

468

but I have not been able to understand how.

The direction that I proceeded was:
Jack gets gold or jack gets silver or jack gets bronze
=$$13*12 + 13*12 + 13*12$$

This seems to be wrong,so if somebody could explain where I'm going wrong and how to arrive at the correct answer it would be great!

Since Jack has to be given an award so we need two more individuals out of rest 13 candidates

Those two candidates may be selected in 13C2 = 78 ways

Now we have team of 3 candidates (Jack included) and three persons can be given three different rewards in 3*2*1 = 3! = 6 ways

Total favourable ways = 78*6 = 468

Bunuel The following options may be given to the question (with C being the correct Answer)

A) 2184
B) 546
C) 468
D) 364
E) 156

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Re: In how many ways can 3 people out of 14 people be given 3 prizes(gold,   [#permalink] 17 Mar 2019, 03:19
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