An international rugby team composed of four people must be organized

Question

An international rugby team composed of four people must be organized from among five Turkish players and six Iranian players. If a team must have at least two Iranian players, how many different ways are there to produce the team?

A. 150

B. 265

C. 315

D. 420

E. 900

The official answer is 265.

I solved the problem in the following way:
Selecting 2 players from 6 iranian players: 6C2 * Selecting the rest of the players: 9C2 (9 players remain after selecting the first two)

Can anyone help me with what's wrong with my method? The problem has been posted here before but the thread is not accessible. Any help will be appreciated.

GMATinsight · Answer

Your calculation is flawed as there are some repeated cases  
e.g Iranians {ABCDEF} and Turkish {PQRST}

Case 1: 2 iranian selected are {AB}, now, 9C2 might select {CQ} 
Case 2: 2 iranian selected are {AC}, now, 9C2 might select {BQ}
Now, you can understadn that these two cases are identical but the calculation has counted it two times hence the REPETITION

You must make separate cases in this question as follows

5 Turkish players and 6 Iranian players, team must have at least 2 Iranian players
Case 1: 2 Iranians and 2 Turkish : 6C2*5C2 = 15*10 = 150
Case 2: 3 Iranians and 1 Turkish : 6C3*5C1 = 20*5 = 100
Case 3: 4 Iranians and 0 Turkish : 6C4*5C0 = 15*1 = 15

Total Outcomes = 150+100+15 = 265

Orvil

Kushchokhani · Answer

Orvil

Your approach is incorrect, leading to wrong answer. We need to make 3 cases as suggested by GMATinsight for solving this particular ques.

pielkay · Answer

Let the Iranian players  = I , and the Turkish players  = T . Analyzing the question, we see that there is a constraint given where each selection of a team of four players must have at least two Iranian players. Therefore, there are three cases that we need to consider to solve this problem, namely:

Case 1 : Two Iranian players and two Turkish players, i.e.  IITT , or  6C2*5C2=150 
OR
 Case 2 : Three Iranian players and one Turkish player, i.e.  IIIT , or  6C3*5C1=100 
OR
 Case 3 : Four Iranian players and no Turkish players, i.e.  IIII , or  6C4=15

Either of these three cases can be true, so we can add them up to get  150+100+15=265 .  Option (B)  is the correct answer.

Hope this helps!

Pritam

ashishLX · Answer

Another approach to answer this could be the reverse way of thinking. 
The question asks for total number of combinations possible when at least 2 players are Iranian. There are two approaches to solve this problem.

Straight forward way 
Here we calculate the possible combinations of these independent events, and eventually add them, since keyword "or" is involved (i.e. mutually exclusive) 
2 Iranian and 2 Turkish = 6C2 * 5C2 = (3*5) * (5*2) = 15*10 = 150
3 Iranian and 1 Turkish = 6C3 * 5C1 = (5*4) * 5 = 20*5 = 100
4 Iranian and 0 Turkish = 6C4 * 5C0 = (5*3) * 1 = 15

Now we add them all up all the likely scenarios for solution => 150 + 100 + 15 = 265

Alternate Approach 
Here we calculate all the possible combinations and deduct all the combinations which cannot happen. 
Total number of combinations = 11C4 = 330
0 Iranian and 4 Turkish = 6C0 * 5C4 = 1*5 = 5
1 Iranian and 3 Turkish = 6C1 * 5C3 = 6*10 = 60

Total number of selection possible - Not getting what we want => 330 - (5+60) = 265

prakashb2497 · Answer

Hi,

I created similar cases like above. However, I didn't use the selection formula and use the basic counting principle to fill in the spots and got a different answer.

Case 1: 2 Iranians and 2 Turkish : 6* 5* 5T*4T = 600
Case 2: 3 Iranians and 1 Turkish : 6*5*4* 5T = 600
Case 3: 4 Iranians and 0 Turkish : 6*5*4*3 = 240

Can you help me understand the flaw in reasoning? Any resources/links to fill in this gap would be really helpful too.

chetan2u   Bunuel   KarishmaB

KarishmaB · Answer

Basic Counting Principle is used when you are selecting from a single group and you need to arrange the selection too. Here we only need to select a group of 4 people from two groups - Iranians and Turkish. There is no arrangement involved. We cannot use the Basic Counting Principle here.
If the 4 needed to be arranged in 4 distinct spots on the team (such as Quarterback, centre, wings and fullback), then we would have arranged the 4 selected players.

oaa3wf · Answer

According to gemini, and this makes sense to me:

Why you can't use the basic counting principle
The basic counting principle (or multiplication principle) is used for a series of  independent  choices, where you multiply the number of options for each step. You cannot use it directly here because the choices are  not independent  due to the constraint that there must be "at least 2 Iranian players." 
A simple multiplication would not account for all the specific sub-conditions of the problem. If you tried to calculate the combinations as a single step—for example, by multiplying the total number of options for each position—you would overcount certain arrangements. For instance, you could not simply choose players one by one, because a Turkish player being chosen for one spot affects the number of remaining spots that must be filled by Iranian players to meet the "at least 2" condition. Instead, the problem must be broken down into discrete, mutually exclusive cases (2 Iranian, 3 Iranian, and 4 Iranian players).

An international rugby team composed of four people must be organized

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An international rugby team composed of four people must be organized

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