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

Question

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

WY919 · Accepted Answer

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!

Bunuel · Answer

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.

mitrakaushi · Answer

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?

D3N0 · Answer

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??

GyMrAT · Answer

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.

Thanks,
GyM

JeffTargetTestPrep · Answer

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

teaserbae · Answer

Bunuel  why we are not taking case 3 where Technology stock is 5, energy is 4 and restaurant is 2
majority will be technology

Bunuel · Answer

Majority means more than half, so majority of 11 is more than 5.5, so 6 or more.

Kinshook · Answer

mitrakaushi 
Given: A portfolio of 11 stocks will be assembled from 7 technology stocks, 6 energy stocks and 2 restaurant stocks.
Asked: How many portfolios are possible if the majority must be technology stocks

Majority mean >=6 stocks must be technology stocks
Remainaing 11-6<=5 may be energy and restaurant stocks

Let technology, energy and restaurants be t, e & r respecitively
t > = 6

(t,e,r) = {(6,3,2),(6,4,1),(6,5,0),(7,3,1),(7,2,2)}
Number of portfolios possible = 7C6*6C3*2C2 + 7C6*6C4*2C1 + 7C6*6C5*2C0 + 7C7*6C3*2C1 + 7C7*6C2*2C2 + 7C7*6C4*2C0
=  7*20*1 + 7*15*2 + 7*6*1 + 1*20*2 + 1*15*1 + 1*15*1 =  140 + 210 + 42 + 40 + 15 + 15 = 462

IMO B

adig244 · Answer

You are counting some cases doubly, for example the tech stocks are from a to g. Selecting 6, let's say a to f, and then you select g along with some other 4 stocks. Now in the case when you select a,b,c,d,e,g for the first 6 tech stocks, the case where you select f and the same combination of 4 other stocks would give the same net combination, counted twice.

Maxsparrow · Answer

I think the reason would be double count.
In this case we account the 7th Tech stock twice in the combinations calculation
1. 7C6
2. 9C5 (9 include 1 Tech stok)

That is why in Bunuel answer, he divided into 2 cases:
1. 6 Tech Stock:
7C6 * 8C5 = 392
2. 7 Tech Stock:
7C7 * 8C4 = 70
Total combinations:  462

Dhimanghosh1234 · Answer

Why cant we have the following where everytime Tech stock gets majority:-

7T 4E
6T 5E
7T 3E 1R
7T 2E 2R
6T 4E 1R
6T 3E 2R

Bunuel · Answer

If you calculate the number of ways to get that correctly, you get the same answer. However, solving the question the way explained above is easier.

peaky · Answer

I think the question is quite ambiguous. The majority can be the largest number of shares, which can be 5 Tech, 2 Restaurant and 4 energy stocks.

RoyJuvera · Answer

Why can't we just do 7C6 * 9C5, meaning you need 6 of the seven and the rest can be whatever including the extra tech one

KarishmaB · Answer

We will be double counting in that case.

Case 1: 
Tech stocks selected through 7C6 = T1, T2, T3, T4, T5, T6
5 stocks selected through 9C5 = T7, E1, E2, E3, E4

Case 2:
Tech stocks selected through 7C6 = T2, T3, T4, T5, T6, T7
5 stocks selected through 9C5 = T2, E1, E2, E3, E4

Your method counts these as two different cases, but actually they are the same.

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

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