An international rugby team composed of four people must be organized

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

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

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

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.