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A certain restaurant offers 6 kinds of cheese and 2 kinds of

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A certain restaurant offers 6 kinds of cheese and 2 kinds of [#permalink] New post 27 Sep 2009, 17:38
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A certain restaurant offers 6 kinds of cheese and 2 kinds of fruit for its dessert platter. If each dessert platter contains an equal number of kinds of cheese and kinds of fruit, how many different dessert platters could the restaurant offer?

A. 8
B. 12
C. 15
D. 21
E. 27
[Reveal] Spoiler: OA

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Re: A certain restaurant offers 6 kinds of cheese and 2 kinds of [#permalink] New post 21 Dec 2009, 11:35
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chetan2u wrote:
A certain restaurant offers 6 kinds of cheese and 2 kinds of fruit for its dessert platter. If each dessert platter contains an equal number of kinds of cheese and kinds of fruit, how many different dessert platters could the restaurant offer?

A) 8
B) 12
C) 15
D) 21
E) 27


As dessert platter should contain equal number of kinds of cheese and fruits, desert can contain:

A. 2 kinds of cheese and 2 kinds of fruits --> 6C2*2C2=15*1=15

B. 1 kind of cheese and 1 kind of fruit --> 6C1*2C1=6*2=12

A+B=15+12=27

Answer: E.
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Re: A certain restaurant offers 6 kinds of cheese and 2 kinds of [#permalink] New post 21 Dec 2009, 12:02
Yup same approach Got E as well.
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Re: A certain restaurant offers 6 kinds of cheese and 2 kinds of [#permalink] New post 22 Dec 2009, 10:25
Please add this one to our combinatorics collection questions.

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Re: A certain restaurant offers 6 kinds of cheese and 2 kinds of [#permalink] New post 26 Jan 2010, 12:04
ahirjoy wrote:
Hi All,

Need your help in solving the below problem,

A certain restaurant offers 6 kinds of cheese and 2 kinds of fruits for its dessert platter. If each dessert platter contains an equal number of kinds of cheese and kinds of fruit, how many different desert platters could the restaurant offer?

A:- 8/12/15/21/27


1 of each: 1 fruit * 1 cheese = 2 * 6 = 12 possibilities
2 of each: Well, you only have 2 fruits, so only one choice here. With 6 cheese, and 2 places, where order does not matter, you get 6!(6-2)!2! = 6!/4!2 = 6*5/2 = 15

So 15 + 12 = 27 total possible platters...
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Re: A certain restaurant offers 6 kinds of cheese and 2 kinds of [#permalink] New post 17 Oct 2010, 18:17
E.
In case we serve 1 of each 6C1*2C1=6*2=12
In case we serve 2 of each 6C2*2C2=15*1=15
=> 15+12=27

.
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Re: A certain restaurant offers 6 kinds of cheese and 2 kinds of [#permalink] New post 18 Jul 2011, 23:54
Bunuel wrote:
chetan2u wrote:
A certain restaurant offers 6 kinds of cheese and 2 kinds of fruit for its dessert platter. If each dessert platter contains an equal number of kinds of cheese and kinds of fruit, how many different dessert platters could the restaurant offer?

A) 8
B) 12
C) 15
D) 21
E) 27


As dessert platter should contain equal number of kinds of cheese and fruits, desert can contain:

A. 2 kinds of cheese and 2 kinds of fruits --> 6C2*2C2=15*1=15

B. 1 kind of cheese and 1 kind of fruit --> 6C1*2C1=6*2=12

A+B=15+12=27

Answer: E.



How do we get 6c2 * 2c2?? Can someone break it down ? Do we read * as "AND" 2 cheese can be chosen from 6 cheese in 6c2 ways AND 2fruit can be choosen from 2 in 2c2 ways hence whenver we have AND we multiply???
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Re: A certain restaurant offers 6 kinds of cheese and 2 kinds of [#permalink] New post 14 Oct 2011, 13:58
Answer:
Option E = 27.

Explanation:

For 1 item each of Fruit & Cheese : 2 x 6 = 12

For 2 item each: Both fruit go in the dish. And we have to select 2 cheese out of 6 i.e. 6-C-2 = 6!/(4! x 2!) = 15

Total Menu Items = 15 +12 = 27
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Re: A certain restaurant offers 6 kinds of cheese and 2 kinds of [#permalink] New post 16 Oct 2011, 12:03
1) One of Each, Therefore 2*6

2) Two of Each,But we have two fruits anyway.So that`ll be one. Rest two Cheese variants can be chosen from the 6 types in 6 C 2 ways.
6! / (4! * 2!)

Total of 1 & 2 gives 27.
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Re: A certain restaurant offers 6 kinds of cheese and 2 kinds of [#permalink] New post 19 Oct 2011, 03:29
ahirjoy wrote:
Hi All,

Need your help in solving the below problem,

A certain restaurant offers 6 kinds of cheese and 2 kinds of fruits for its dessert platter. If each dessert platter contains an equal number of kinds of cheese and kinds of fruit, how many different desert platters could the restaurant offer?

A:- 8/12/15/21/27


6C1*2C1 + 6C2*2C2 = 27
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Re: A certain restaurant offers 6 kinds of cheese and 2 kinds of [#permalink] New post 01 Nov 2011, 06:07
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the platter could contain either 2 items or 4 items.
2-items: \(6*2 = 12\)

4-items: \(\frac{6*5}{2!}*\frac{2*1}{2!} = 15\)

total = 15+12 = 27
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Re: A certain restaurant offers 6 kinds of cheese and 2 kinds of [#permalink] New post 01 Nov 2011, 13:20
E

Two possibilities, either 2 types of each or 1 type of each.

For 2 types: 6C2 * 2C2 = 15 * 1 = 15
For 1 type: 6C1 * 2C1 = 6 * 2 = 12

Total is 15+12 = 27.
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Re: A certain restaurant offers 6 kinds of cheese and 2 kinds of [#permalink] New post 09 Feb 2012, 11:32
Expert's post
siddhans wrote:
Bunuel wrote:
chetan2u wrote:
A certain restaurant offers 6 kinds of cheese and 2 kinds of fruit for its dessert platter. If each dessert platter contains an equal number of kinds of cheese and kinds of fruit, how many different dessert platters could the restaurant offer?

A) 8
B) 12
C) 15
D) 21
E) 27


As dessert platter should contain equal number of kinds of cheese and fruits, desert can contain:

A. 2 kinds of cheese and 2 kinds of fruits --> 6C2*2C2=15*1=15

B. 1 kind of cheese and 1 kind of fruit --> 6C1*2C1=6*2=12

A+B=15+12=27

Answer: E.


How do we get 6c2 * 2c2?? Can someone break it down ? Do we read * as "AND" 2 cheese can be chosen from 6 cheese in 6c2 ways AND 2fruit can be choosen from 2 in 2c2 ways hence whenver we have AND we multiply???


Yes, your understanding is correct.

It's called Principle of Multiplication: if one event can occur in m ways and a second can occur independently of the first in n ways, then the two events can occur in m*n ways.
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Re: A certain restaurant offers 6 kinds of cheese and 2 kinds of [#permalink] New post 10 Feb 2012, 09:08
6+6+6c2=27.
E is the answer .
The above expression denoted choosing one cheese type and one fruit type (6+6) ways ,for each type of fruit there are 6 types of cheese that can be selected .
And finally where there are two types of cheese and fruits respectively .
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Re: A certain restaurant offers 6 kinds of cheese and 2 kinds of [#permalink] New post 15 Feb 2012, 10:52
I got e as well

For 2 types: 6C2 * 2C2 = 15 * 1 = 15
For 1 type: 6C1 * 2C1 = 6 * 2 = 12

exact approach
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Re: A certain restaurant offers 6 kinds of cheese and 2 kinds of [#permalink] New post 15 Aug 2014, 23:54
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Re: A certain restaurant offers 6 kinds of cheese and 2 kinds of [#permalink] New post 26 May 2015, 22:24
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Re: A certain restaurant offers 6 kinds of cheese and 2 kinds of [#permalink] New post 27 May 2015, 00:50
Expert's post
This question is a good example where both OR and AND events are used. Let's see how such events are treated when finding the number of ways of an event happening.

Given Info
We are given that a restaurant in its dessert platter offers 6 kinds of cheese and 2 kinds of fruits. We are asked to find the number of ways a dessert platter can be offered if the platter contains equal kinds of fruits and cheese.

Approach
We observe here that there are lesser kinds of fruits than the cheese. Hence the number of ways the platter can be offered is limited by the number of fruits.

Since a platter would be incomplete without any of the fruit or cheese, for the event to occur( i.e. platter to be served) a platter should have fruit AND cheese.

Also, there can be a case where a platter may have 1 fruit AND 1 cheese OR 2 fruits AND 2 cheese. Hence, the event(i.e. platter to be served) can occur in either of the ways.

Number of ways the platter can be served = Number of ways of picking up 1 fruit out of 2 fruits AND number of ways of picking up 1 cheese out of 6 cheese OR Number of ways of picking up 2 fruits out of 2 fruits AND number of ways of picking up 2 cheese out of 6 cheese

Working Out
Number of ways in which 1 fruit can be picked out of 2 fruits = \(2_{c_{1}}\)

Number of ways in which 1 cheese can be picked out of 6 cheese = \(6_{c_{1}}\)

Hence number of ways in which a platter having 1 fruit and 1 cheese can be served = \(2_{c_{1}}\) \(*\) \(6_{c_{1}}\) \(= 2 * 6 = 12\)

Similarly, Number of ways in which 2 fruits can be picked out of 2 fruits = \(2_{c_{2}}\)

Number of ways in which 2 cheese can be picked out of 6 cheese = \(6_{c_{2}}\)

Hence number of ways in which a platter having 2 fruits and 2 cheese can be served = \(2_{c_{2}}\) \(*\) \(6_{c_{2}}\) \(= 1 * 15 = 15\)

Therefore total number of ways of serving a dessert platter \(= 12 + 15 = 27\)

Hope this helps :)

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Re: A certain restaurant offers 6 kinds of cheese and 2 kinds of [#permalink] New post 27 May 2015, 04:20
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A certain restaurant offers 6 kinds of cheese and 2 kinds of fruit for its dessert platter. If each dessert platter contains an equal number of kinds of cheese and kinds of fruit, how many different dessert platters could the restaurant offer?

A) 8
B) 12
C) 15
D) 21
E) 27

Since the plates have to contain an equal number of cheese and fruit types, the only possibilities are 1 kind of cheese and 1 kind of fruit or 2 types of cheese and 2 types of fruit.

Case 1:
Cheese: n=6, r = 1
(6x5x4x3x2x1)
1(5x4x3x2x1)

Fruit: n=2, r=1
(2x1)
1(1)

Combining cheese and fruit 6 x 2 = 12

Case 2:
Cheese n = 6, r = 2
(6x5x4x3x2x1)
2x1 (4x3x2x1)
=15

Fruit: There are only two fruits, so there is only one combination.

Combining cheese and fruits 15 x 1 = 15

15 1fruit-1cheese combinations + 12 2fruit-2cheese combinations = 27 total combinations
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Re: A certain restaurant offers 6 kinds of cheese and 2 kinds of   [#permalink] 27 May 2015, 04:20
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