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Six friends live in the city of Monrovia. There are four nat

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Six friends live in the city of Monrovia. There are four nat [#permalink]

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Found this question in one of the Veritas Prep blog posts...

Six friends live in the city of Monrovia. There are four natural attractions around Monrovia – a waterfall, a safari, a lake and some caves. The friends decide to take a vacation together at one of these attractions. To select the attraction, each one of them votes for one of the attractions. What is the probability that each attraction gets at least one vote?

[Reveal] Spoiler: OE
The answer is 1560/(4^6). I was wondering if we can get to this answer by removing the cases where [an "attraction" gets zero votes] can be subtracted from the [total number of cases].
[Reveal] Spoiler: OA

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Re: Six friends live in the city of Monrovia. There are four nat [#permalink]

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v1gnesh wrote:
Found this question in one of the Veritas Prep blog posts...

Six friends live in the city of Monrovia. There are four natural attractions around Monrovia – a waterfall, a safari, a lake and some caves. The friends decide to take a vacation together at one of these attractions. To select the attraction, each one of them votes for one of the attractions. What is the probability that each attraction gets at least one vote?

[Reveal] Spoiler: OE
The answer is 1560/(4^6). I was wondering if we can get to this answer by removing the cases where [an "attraction" gets zero votes] can be subtracted from the [total number of cases].


The cases when attractions get zero votes are:

{6, 0, 0, 0}
{5, 1, 0, 0}
{4, 2, 0, 0}
{4, 1, 1, 0}
{3, 2, 1, 0}
{3, 1, 1, 1}
{2, 2, 2, 0}
{2, 2, 1, 1}

Counting those is harder than a direct approach when we need to count only:
{3, 1, 1, 1} --> \(\frac{4!}{3!}*C^3_6*3!=480\).
{2, 2, 1, 1} --> \(\frac{4!}{2!2!}*C^2_6*C^2_4*C^1_2=1,080\).

The total number of ways in which the 6 friends can vote = 4*4*4*4*4*4 = 4^6.

P = (480 + 1,080)/4^6 = 1560/4^6.

Hope it helps.
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Re: Six friends live in the city of Monrovia. There are four nat [#permalink]

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New post 28 Apr 2014, 05:35
Bunuel wrote:
v1gnesh wrote:
Found this question in one of the Veritas Prep blog posts...

Six friends live in the city of Monrovia. There are four natural attractions around Monrovia – a waterfall, a safari, a lake and some caves. The friends decide to take a vacation together at one of these attractions. To select the attraction, each one of them votes for one of the attractions. What is the probability that each attraction gets at least one vote?

[Reveal] Spoiler: OE
The answer is 1560/(4^6). I was wondering if we can get to this answer by removing the cases where [an "attraction" gets zero votes] can be subtracted from the [total number of cases].


The cases when attractions get zero votes are:

{6, 0, 0, 0}
{5, 1, 0, 0}
{4, 2, 0, 0}
{4, 1, 1, 0}
{3, 2, 1, 0}
{3, 1, 1, 1}
{2, 2, 2, 0}
{2, 2, 1, 1}

Counting those is harder than a direct approach when we need to count only:
{3, 1, 1, 1} --> \(\frac{4!}{3!}*C^3_6*3!=480\).
{2, 2, 1, 1} --> \(\frac{4!}{2!2!}*C^2_6*C^2_4*C^1_2=1,080\).

The total number of ways in which the 6 friends can vote = 4*4*4*4*4*4 = 4^6.

P = (480 + 1,080)/4^6 = 1560/4^6.

Hope it helps.


Bunuel

Please Elaborate how you got the 3 Equations :

Counting those is harder than a direct approach when we need to count only:
{3, 1, 1, 1} --> \(\frac{4!}{3!}*C^3_6*3!=480\).
{2, 2, 1, 1} --> \(\frac{4!}{2!2!}*C^2_6*C^2_4*C^1_2=1,080\).

The total number of ways in which the 6 friends can vote = 4*4*4*4*4*4 = 4^6.

Thankyou!

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Re: Six friends live in the city of Monrovia. There are four nat [#permalink]

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New post 28 Apr 2014, 10:11
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niyantg wrote:
Bunuel wrote:
v1gnesh wrote:
Found this question in one of the Veritas Prep blog posts...

Six friends live in the city of Monrovia. There are four natural attractions around Monrovia – a waterfall, a safari, a lake and some caves. The friends decide to take a vacation together at one of these attractions. To select the attraction, each one of them votes for one of the attractions. What is the probability that each attraction gets at least one vote?

[Reveal] Spoiler: OE
The answer is 1560/(4^6). I was wondering if we can get to this answer by removing the cases where [an "attraction" gets zero votes] can be subtracted from the [total number of cases].


The cases when attractions get zero votes are:

{6, 0, 0, 0}
{5, 1, 0, 0}
{4, 2, 0, 0}
{4, 1, 1, 0}
{3, 2, 1, 0}
{3, 1, 1, 1}
{2, 2, 2, 0}
{2, 2, 1, 1}

Counting those is harder than a direct approach when we need to count only:
{3, 1, 1, 1} --> \(\frac{4!}{3!}*C^3_6*3!=480\).
{2, 2, 1, 1} --> \(\frac{4!}{2!2!}*C^2_6*C^2_4*C^1_2=1,080\).

The total number of ways in which the 6 friends can vote = 4*4*4*4*4*4 = 4^6.

P = (480 + 1,080)/4^6 = 1560/4^6.

Hope it helps.


Bunuel

Please Elaborate how you got the 3 Equations :

Counting those is harder than a direct approach when we need to count only:
{3, 1, 1, 1} --> \(\frac{4!}{3!}*C^3_6*3!=480\).
{2, 2, 1, 1} --> \(\frac{4!}{2!2!}*C^2_6*C^2_4*C^1_2=1,080\).

The total number of ways in which the 6 friends can vote = 4*4*4*4*4*4 = 4^6.

Thankyou!


For each attraction to get at least one vote we should have the following distributions of 6 votes: {3, 1, 1, 1} or {2, 2, 1, 1}.

A - B - C - D (attractions)
{3, 1, 1, 1} --> \(\frac{4!}{3!}*C^3_6*3!=480\), where:
\(\frac{4!}{3!}\) is the # assignments of {3, 1, 1, 1} to attractions (A gets 3, B gets 1, C gets 1, D gets 1; A gets 1, B gets 3, C gets 1, D gets 1; A gets 1, B gets 1, C gets 3, D gets 1; A gets 1, B gets 1, C gets 1, D gets 3);
\(C^3_6\) is selecting the 3 people who will vote for the attraction with 3 votes;
3! is the distribution of other 3 votes among the remaining attractions.

The same logic applies to {2, 2, 1, 1} case.

As for 4^6: each friend can vote in 4 ways (for waterfall, safari, lake or caves). So, the total number of ways in which the 6 friends can vote = 4*4*4*4*4*4 = 4^6.

Hope it's clear.
_________________

New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics

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Re: Six friends live in the city of Monrovia. There are four nat [#permalink]

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Re: Six friends live in the city of Monrovia. There are four nat   [#permalink] 03 Jan 2016, 04:39
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