M24-16

we have 6 boys. need to form two groups of 3 each.

so we need to see how many ways we can choose 3 from these 6 = 6!/3!*3! = 20 ways

so we can actually form 20 sets of 3 from these 6

and for the last part as bunel explained, in a group (a,b,c,d,e,f) - what we are doing in the above formula is finding ways in which groups of three can be made from these 6. so eg (a,b,c) will be one of those 20 and (d,e,f) will also be one of that 20.

now think about this in our scenario only 2 teams are formed. so if (a,b,c) are 1 team automatically the rest go to another team. so there is no need to count(d,e,f). same way if (d,e,f) is 1 team the rest are another team. so we can see that 20 has double counts. so we divide by 2. so ans is 10.

Hello,
I'm very much confused about when to consider the order and when not to. I made two similar mistakes in this CAT which did affect the end result to 50. Please help me understand the order in permutation and combination questions.

Combinatorics Made Easy!

Theory on Combinations

DS questions on Combinations
PS questions on Combinations

Tough and tricky questions on Combinations

Probability Made Easy!

Theory on probability problems

Data Sufficiency Questions on Probability
Problem Solving Questions on Probability

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Hope it helps.

Hi Bunnel ,i have not understood the 3C3 part . Why do we need to multiply this 3C3 with 6C3/2! , Can you explain please .

To make it simpler, just imagine you were doing a game of 2v2. 4 dudes - Al, Ben, Chuck, Dan. How many ways can you choose two teams of two? (4c2)*(2c2)=6. You can only get {Al+Ben v Chuck+Dan}, {Al+Chuck v Ben+Dan}, and {Al+Dan v Ben+Chuck}. But wait! Because order does not matter, {Al+Ben v Chuck+Dan} is the same as {Chuck+Dan v Al+Ben} and you actually need to divide the 6 possibilities by the number of groups - 2. Using math, (4c2)*(2c2)/2! = 3.

In this problem, you have 6 dudes - Al, Ben, Chuck, Dan, Evan, Frank. Let's turn this into a game of 3 on 3.

Okay, you need to choose 3 for one of the teams. 6c3=20. So, there are 20 ways to pick 3 dudes from a group of 6. Let's call this first group Group A.

You have one team now and need to "create" another team. How many ways can you chose 3 dudes from the remaining 3 dudes? 3c3=1 way. Let's call this second group Group B.

Now, think about the 20 ways you could pick 2 groups of 3 dudes from 6. If one team consists of Al, Ben, and Chuck in Group A, this is one of the 20 possible groups for Group A. This means that Group B (the other team) would have to be Dan, Evan, and Frank. [b] However, let's think about the other 19 possibilities. Picking Dan, Evan, and Frank in Group A is one of those other 19 possibilities, and one of the total of 20 possibilities. This would matter if the order mattered, but because you just want two teams - you don't really care which team is Group A and which is Group B -, the order does not matter and these 20 choices actually have duplicates. So, divide 20 by 2 to get 10.

In math speak, (6c3)*(3c3)/2!=10.

Hi,

you could also use the Basic counting principle.

For the first spot in group 1 you have 6 possibilities (6 boys)
for the seconde 5 possibilities
for the third 4 poss..
for the first in the second group 3 possi.
for the second in the second group 2

So in total 6x5x4x3x2x1

Because the Order in the group doesn't matter (e.g. for Group 1 {A,B,C} = {B,C,A}) you have to divide bei 3!
The Order of the Groups also doesn't matter ( G1, G2 = G2, G1) so you have to divide bei 2! again

in Conclusion:

\(\frac{6x5x4x3x2x1}{3!x2!}\)

Cheers

I think this is a high-quality question and I agree with explanation.

I think this is a high-quality question and I agree with explanation.

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

Bunuel

When do we know that the order of the teams does matter? Is there a type of word(s) in the stem that indicates that the order of teams does matter?

Thanks in advance!

­I think this post might help:

https://gmatclub.com/forum/in-how-many- ... ml#p642632

Bunuel, Won't the composition - Team 1 : A,B,C and Team 2 : D,E,F and the composition - Team 1 : D,E,F and Team 2: A,B,C be considered different teams? This is why I did not multiply by 2!. Can you please help me understand this, thank you.

No, they are not considered different. Because the problem states “two teams of three” without labeling them (e.g., Team 1 vs. Team 2), the pair {A,B,C} & {D,E,F} is the same as {D,E,F} & {A,B,C}. We divide by 2! to avoid counting these identical partitions twice.

Bunuel, understood, thank you!