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A portfolio of 11 stocks will be assembled from 7 technology stocks, 6

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A portfolio of 11 stocks will be assembled from 7 technology stocks, 6  [#permalink]

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New post Updated on: 05 May 2018, 00:12
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A portfolio of 11 stocks will be assembled from 7 technology stocks, 6 energy stocks and 2 restaurant stocks. How many portfolios are possible if the majority must be technology stocks

A. 231
B. 462
C. 693
D. 924
E. 1155

Originally posted by mitrakaushi on 04 May 2018, 19:07.
Last edited by Bunuel on 05 May 2018, 00:12, edited 1 time in total.
Renamed the topic.
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Re: A portfolio of 11 stocks will be assembled from 7 technology stocks, 6  [#permalink]

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New post 04 May 2018, 19:12
Why can't we employ the simple strategy as follows:

Min tech stocks needed: 6

Number of ways they can be selected: 7C6 = 7

Stocks left (Including the left out tech stock) : 9

To complete the number 11, we need to find 5 more stocks: 9C5 = 126

So Total Number: 126 * 9 = 1134.

My calculation is wrong, but I can't figure out the flaw in the logic. Can someone help me with this?
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Re: A portfolio of 11 stocks will be assembled from 7 technology stocks, 6  [#permalink]

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New post 04 May 2018, 20:31
Here is my way:

Majority must be tech stock so chose 6 or 7 from 7 tech stocks ----7C6 and 7C7

then chose the rest from the other 8 stocks

So we get 7c6 x 8c5 + 7c7 x 8c4 = 7x56 + 70 =462

Cheers!
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Re: A portfolio of 11 stocks will be assembled from 7 technology stocks, 6  [#permalink]

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New post 04 May 2018, 23:16
WY919 wrote:
Here is my way:

Majority must be tech stock so chose 6 or 7 from 7 tech stocks ----7C6 and 7C7

then chose the rest from the other 8 stocks

So we get 7c6 x 8c5 + 7c7 x 8c4 = 7x56 + 70 =462

Cheers!


My Confusion/doubt....
What if we take 5 Tech Stock, 4 out of 6 and 2 out of 2. it still gives us majority of Tech stock.
So what does majority actually means here?? is it means greater then all other stock combined or greater than any other individual type of tock in the set??
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Re: A portfolio of 11 stocks will be assembled from 7 technology stocks, 6  [#permalink]

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New post 05 May 2018, 00:33
mitrakaushi wrote:
A portfolio of 11 stocks will be assembled from 7 technology stocks, 6 energy stocks and 2 restaurant stocks. How many portfolios are possible if the majority must be technology stocks

A. 231
B. 462
C. 693
D. 924
E. 1155


In a portfolio of 11 stocks the majority to be technology stocks, the number of technology stocks must be more than or equal to 6. Since there are only 7 technology stocks available then there should be 6 or 7 technology stocks in the portfolio of 11 stocks.


Case 1: 6 technology stocks.

6 technology stocks, any 5 stocks from 6 + 2 = 8 stocks
7C6*8C5 = 7*56 = 392.


Case 2: 7 technology stocks.

7 technology stocks, any 4 stocks from 6 + 2 = 8 stocks.
7C7*8C4 = 1*70 = 70.


Total = 392 + 70 = 462.


Answer: B.
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Re: A portfolio of 11 stocks will be assembled from 7 technology stocks, 6  [#permalink]

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New post 05 May 2018, 09:34
Total stocks:
7 Technology stocks
6 Energy Stocks
2 Restaurant Stocks

Total Stocks needed in portfolio = 11 stocks

Constraint: Majority of stocks in portfolio should be technology stocks.

So technology shares should be 6 or 7.

1. If the technology stock is 6 - then other stocks should be 5
7C6 * 8C5 = 7 * 56 = 392

2. If the technology stock is 7 - then other stocks should be 4
7C7 * 8C4 = 1 * 70 = 70

Total = 70 + 392 = 462

Ans: B
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Re: A portfolio of 11 stocks will be assembled from 7 technology stocks, 6  [#permalink]

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New post 05 May 2018, 13:13
WY919 wrote:
Here is my way:

Majority must be tech stock so chose 6 or 7 from 7 tech stocks ----7C6 and 7C7

then chose the rest from the other 8 stocks

So we get 7c6 x 8c5 + 7c7 x 8c4 = 7x56 + 70 =462

Cheers!



++++++++++++++++++++++++++++++++++++++++

I can choose 5 + 4 + 2 = 11 as well. In this case the tech stocks are still in the majority
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Re: A portfolio of 11 stocks will be assembled from 7 technology stocks, 6  [#permalink]

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New post 05 May 2018, 13:46
i think there should be 3 cases. The two cases as mentioned in previous replies while third one mentioned below.

Case 3: 5 Technology stocks, 4 Energy Stocks, 2 Restaurant stocks.
Still technology stocks are in majority.
so, 7C5 * 6C4 * 2C2 = 315

Total = 70 +392+315 = 777 (which is not in the option though)

Can anyone tell what is mistake in this particular case.
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Re: A portfolio of 11 stocks will be assembled from 7 technology stocks, 6  [#permalink]

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New post 05 May 2018, 13:53
nightvision

The third case you mentioned (5 Technology stocks, 4 Energy Stocks, 2 Restaurant stocks) is not included because in this case the technology shares might be higher than the individual Energy and Restaurant stocks , but not the majority in the portfolio as whole (5 out of 11)

The portfolio should consists of 11 shares out of which 5 technology shares (as in case 3) might not be major part of portfolio. Technology shares should be 6 or 7 [In this 7 is maximum available] to have tech shares as majority.
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Re: A portfolio of 11 stocks will be assembled from 7 technology stocks, 6  [#permalink]

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New post 06 May 2018, 10:14
Bunuel wrote:
mitrakaushi wrote:
A portfolio of 11 stocks will be assembled from 7 technology stocks, 6 energy stocks and 2 restaurant stocks. How many portfolios are possible if the majority must be technology stocks

A. 231
B. 462
C. 693
D. 924
E. 1155


In a portfolio of 11 stocks the majority to be technology stocks, the number of technology stocks must be more than or equal to 6. Since there are only 7 technology stocks available then there should be 6 or 7 technology stocks in the portfolio of 11 stocks.


Case 1: 6 technology stocks.

6 technology stocks, any 5 stocks from 6 + 2 = 8 stocks
7C6*8C5 = 7*56 = 392.


Case 2: 7 technology stocks.

7 technology stocks, any 4 stocks from 6 + 2 = 8 stocks.
7C7*8C4 = 1*70 = 70.


Total = 392 + 70 = 462.


Answer: B.


Bunuel

Why can't we employ the simple strategy as follows:

Min tech stocks needed: 6

Number of ways they can be selected: 7C6 = 7

Stocks left (Including the left out tech stock) : 9

To complete the number 11, we need to find 5 more stocks: 9C5 = 126

So Total Number: 126 * 9 = 1134.

My calculation is wrong, but I can't figure out the flaw in the logic. Can someone help me with this?
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Re: A portfolio of 11 stocks will be assembled from 7 technology stocks, 6  [#permalink]

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New post 06 May 2018, 13:26
I did the way I listed below, but it's a long solution and prone to errors. Use the solution bunnel posted above. It's much quicker.

My method:
Technology stocks must be majority per question stem. This can be true when there are either 6 technology stocks or 7 (why? because it must be greater than 11/2 = 5.5 and there are only 7 tech stocks available).

Tech stocks = 6:
nCr(7,6) * nCr(6,5) + nCr(7,6) * nCr(6,4) * nCr(2,1) + nCr(7,6) * nCr(6,3) * nCr(2,2)

This sums to 392

Tech stocks = 7:
nCr(7,7) * nCr(6,4) + nCr(7,7) * nCr(6,3) * nCr(2,1) + nCr(7,7) * nCr(6,2) * nCr(2,2)

This sums to 70

392 + 70 = 462.

Answer = B.

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Re: A portfolio of 11 stocks will be assembled from 7 technology stocks, 6  [#permalink]

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New post 06 May 2018, 13:53
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mitrakaushi

As per me first of all as per your approach total should be =7*126=882

Next flaw in your approach
I would say u have already selected tech stocks 6 out of 7 then you included the left over among other and start selecting from them
Whereas principle of selection says if one thing is selected then it should not be included in remaining items that's why u r getting wrong result

Make 2 cases as bunuel mentioned above then you will surely get correct answer

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Re: A portfolio of 11 stocks will be assembled from 7 technology stocks, 6  [#permalink]

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New post 07 May 2018, 03:13
Bunuel wrote:
mitrakaushi wrote:
A portfolio of 11 stocks will be assembled from 7 technology stocks, 6 energy stocks and 2 restaurant stocks. How many portfolios are possible if the majority must be technology stocks

A. 231
B. 462
C. 693
D. 924
E. 1155


In a portfolio of 11 stocks the majority to be technology stocks, the number of technology stocks must be more than or equal to 6. Since there are only 7 technology stocks available then there should be 6 or 7 technology stocks in the portfolio of 11 stocks.


Case 1: 6 technology stocks.



6 technology stocks, any 5 stocks from 6 + 2 = 8 stocks
7C6*8C5 = 7*56 = 392.


Case 2: 7 technology stocks.

7 technology stocks, any 4 stocks from 6 + 2 = 8 stocks.
7C7*8C4 = 1*70 = 70.


Total = 392 + 70 = 462.


Answer: B.


why can't we take the case where 5 are tech stocks, 4 are energy stocks, and 2 are restaurant stocks.
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A portfolio of 11 stocks will be assembled from 7 technology stocks, 6  [#permalink]

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New post 07 May 2018, 12:25
thefibonacci wrote:
Bunuel wrote:
mitrakaushi wrote:
A portfolio of 11 stocks will be assembled from 7 technology stocks, 6 energy stocks and 2 restaurant stocks. How many portfolios are possible if the majority must be technology stocks

A. 231
B. 462
C. 693
D. 924
E. 1155


In a portfolio of 11 stocks the majority to be technology stocks, the number of technology stocks must be more than or equal to 6. Since there are only 7 technology stocks available then there should be 6 or 7 technology stocks in the portfolio of 11 stocks.


Case 1: 6 technology stocks.



6 technology stocks, any 5 stocks from 6 + 2 = 8 stocks
7C6*8C5 = 7*56 = 392.


Case 2: 7 technology stocks.

7 technology stocks, any 4 stocks from 6 + 2 = 8 stocks.
7C7*8C4 = 1*70 = 70.


Total = 392 + 70 = 462.


Answer: B.


why can't we take the case where 5 are tech stocks, 4 are energy stocks, and 2 are restaurant stocks.


Because question talks about being majority of technology stocks.
In your case other than tech becomes 6 which is majority as compared to 5 of tech.
So that case is not relevant here.

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Re: A portfolio of 11 stocks will be assembled from 7 technology stocks, 6  [#permalink]

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New post 13 Jun 2018, 08:32
Since the majority of the stocks should be tech companies then this could be achieved by (7C7 * 8C4) + (7C6*8C5) = 462 ways.
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Re: A portfolio of 11 stocks will be assembled from 7 technology stocks, 6  [#permalink]

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New post 13 Jun 2018, 23:52
11 stocks to be chosen from 7 technology stocks, 6 energy stocks and 2 restaurant stocks.
Condition: the majority must be technology stocks.

For technology stocks to be in majority of 11 stocks, the split should be ( 6, 5 ) or ( 7, 4 ).

Case 1: ( 6, 5 )
Choosing 6 technology stocks can be done in 7C6 ways.
Remaining 5 stocks should be chosen from 6 + 2 stocks. This can be done in 8C5 ways.
Total = 7C6 X 8C5.

Case 2: ( 7, 4 )
Choosing 7 technology stocks can be done in 7C7 ways.
Remaining 4 stocks should be chosen from 6 + 2 stocks. This can be done in 8C4 ways.
Total = 7C7 X 8C4.

Total = 7C6 X 8C5 + 7C7 X 8C4 = 462.
Option B
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Re: A portfolio of 11 stocks will be assembled from 7 technology stocks, 6  [#permalink]

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New post 14 Jun 2018, 14:12
mitrakaushi wrote:
A portfolio of 11 stocks will be assembled from 7 technology stocks, 6 energy stocks and 2 restaurant stocks. How many portfolios are possible if the majority must be technology stocks

A. 231
B. 462
C. 693
D. 924
E. 1155


If the majority of the stock has to be from technology, then either atleast 6 technology stocks or 7 technology stocks should be chosen.

(i) 6 tech stocks can be selected in \(7C6\) ways & the balance 5 stocks can be selected in \(8C5\) ways

hence # of ways if 6 tech stocks are selected = \(7C6 * 8C5\) = 392

(ii) 7 tech stocks can be selected in \(7C7\) ways & the balance 4 stocks can be selected in \(8C4\) ways

hence # of ways if 7 tech stocks are selected = \(7C7 * 8C4\) = 70

Total # of ways = 392 + 70 = 462

Answer B.


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Re: A portfolio of 11 stocks will be assembled from 7 technology stocks, 6  [#permalink]

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New post 17 Jun 2018, 19:41
mitrakaushi wrote:
A portfolio of 11 stocks will be assembled from 7 technology stocks, 6 energy stocks and 2 restaurant stocks. How many portfolios are possible if the majority must be technology stocks

A. 231
B. 462
C. 693
D. 924
E. 1155


If technology stocks are to have majority, either 7 or 6 of the stocks in the portfolio must be technology stocks.

When there are 7 technology stocks in the portfolio, we have 8 stocks left for 4 spots, which can be chosen in 8C4 = (8 x 7 x 6 x 5)/4! = = (8 x 7 x 6 x 5)/(4 x 3 x 2) = 70 ways.

When there are 6 technology stocks in the portfolio, we have 5 stocks left for 5 spots, which can be chosen in 8C5 = (8 x 7 x 6 x 5 x 4)/5! = = (8 x 7 x 6 x 5 x 4)/(5 x 4 x 3 x 2) = 56 ways.
Furthermore, 6 technology stocks can be chosen from a total of 7 technology stocks in 7C6 = 7 ways.
Thus, in total, there are 7 x 56 = 392 ways to have 6 technology stocks in the portfolio.

So are a total of 70 + 392 = 462 portfolios where the technology stocks have majority.

Answer: B
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Re: A portfolio of 11 stocks will be assembled from 7 technology stocks, 6 &nbs [#permalink] 17 Jun 2018, 19:41
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