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In how many different ways can a group of 8 people be divided into 4 teams of 2 people each?
A. 90 B. 105 C. 168 D. 420 E. 2520
The solution to this problem is the number of combinations. First we get one team out of 8 . The number of ways to do this would be \(C_8^2\). The next combination is 2 out of 6 or \(C_6^2\), and so on. Having all four combinations multiplied, we need to divide the total number by the number of ways the teams can be chosen \(4!\), since we are not interested if the team with two certain people is chosen first, second or third. Therefore, the answer is found by the following formula \(\frac{C_8^2*C_6^2*C_4^2*C_2^2}{4!}=105\).
If the question were"order doesn't matter", would the answer be 2C8/4!?
Please explain. What is the issue of order here?
In how many different ways can a group of 8 people be divided into 4 teams of 2 people each?
A. 90 B. 105 C. 168 D. 420 E. 2520
\(\frac{C^2_8*C^2_6*C^2_4*C^2_2}{4!}=105\), we are dividing by 4! (factorial of the # of teams) as the order of the teams does not matter. If 8 people are - 1, 2, 3, 4, 5, 6, 7, 8, then (1,2)(3,4)(5,6)(7,8) would be the same 4 teams as (5,6)(7,8)(1,2)(3,4), as we don't have team #1, team #2...
You can think about this in another way. For the first person we can pick a pair in 7 ways; For the second one in 5 ways (as two are already chosen); For the third one in 3 ways (as 4 people are already chosen); For the fourth one there is only one left.
okay i didn't really get it and i don't expect anyone to explain . but my Gmat test is on sunday and i would like someone to give me a rule here if possible. this division by 4! reminded me of something we learned at school when studying probability and i remember there was a rule about how to identify questions where you calculate a numerator using a combination and then divide by a factorial , there is a certain type of questions that requires this "unique approach" , does anyone know how to identify this type of questions? i don't want a general explanation about combinations because i know them , just how to identify a question where in the answer you need to divide by a factorial .
In how many different ways can a group of 8 people be divided into 4 teams of 2 people each?
A. 90 B. 105 C. 168 D. 420 E. 2520
The solution to this problem is the number of combinations. First we get one team out of 8 . The number of ways to do this would be \(C_8^2\). The next combination is 2 out of 6 or \(C_6^2\), and so on. Having all four combinations multiplied, we need to divide the total number by the number of ways the teams can be chosen \(4!\), since we are not interested if the team with two certain people is chosen first, second or third. Therefore, the answer is found by the following formula \(\frac{C_8^2*C_6^2*C_4^2*C_2^2}{4!}=105\).
Answer: B
Hi,
I'm having a difficult time conceptualizing the explanation to this problem. I'm not good with formal notation for combination problems so I use the simple "anagram" method, which I learned from MGMAT. If anyone is familiar with that approach, could you please post an explanation to the problem using that language?
In approaching this problem, my thought process was something similar to what follows below.
8 people into 4 teams of 2... Okay 8 choose 2, 4 different ways. So I did an anagram with 1, 2, 3, ...8 and A, A, B, B, ... H, H, something I represented mathematically with 8! / 2!*2!*2!*2!. Then I thought about how many ways I could make 4 teams out of that and divided the result, which was 2520, by 4! and got the answer, 105.
I feel like my approach was very lucky in this situation and won't work well in the future.
Another way to think about this is remove the: - meaning for order within the groups - meaning for order among the groups solution: 1. we arrange all the 8 people in a line, this gives us: 8! 2. since the order among the gourps does not matter, we will have to devide 8! by 4! 3. since the order within the groups allso does not matter, for each group, we will have to devide by 2!
hence we got: 8!/(4!2!2!2!2!) = 105
comment: by saying "the order does not matter" i mean that, for example if we take the order within the gourp, it does matter if i picked person X to be the first in the group and later person Y to be in the group.
In how many different ways can a group of 8 people be divided into 4 teams of 2 people each?
A. 90 B. 105 C. 168 D. 420 E. 2520
The solution to this problem is the number of combinations. First we get one team out of 8 . The number of ways to do this would be \(C_8^2\). The next combination is 2 out of 6 or \(C_6^2\), and so on. Having all four combinations multiplied, we need to divide the total number by the number of ways the teams can be chosen \(4!\), since we are not interested if the team with two certain people is chosen first, second or third. Therefore, the answer is found by the following formula \(\frac{C_8^2*C_6^2*C_4^2*C_2^2}{4!}=105\).
Answer: B
Bunuel, can you please explain why the total no. of ways is divided by 4! when we are not interested if the team with two certain people is chosen first, second or third? I mean how have you determined that it should be divided by 4! ?
In how many different ways can a group of 8 people be divided into 4 teams of 2 people each?
A. 90 B. 105 C. 168 D. 420 E. 2520
The solution to this problem is the number of combinations. First we get one team out of 8 . The number of ways to do this would be \(C_8^2\). The next combination is 2 out of 6 or \(C_6^2\), and so on. Having all four combinations multiplied, we need to divide the total number by the number of ways the teams can be chosen \(4!\), since we are not interested if the team with two certain people is chosen first, second or third. Therefore, the answer is found by the following formula \(\frac{C_8^2*C_6^2*C_4^2*C_2^2}{4!}=105\).
Answer: B
Bunuel, can you please explain why the total no. of ways is divided by 4! when we are not interested if the team with two certain people is chosen first, second or third? I mean how have you determined that it should be divided by 4! ?
In how many different ways can a group of 8 people be divided into 4 teams of 2 people each?
A. 90 B. 105 C. 168 D. 420 E. 2520
The solution to this problem is the number of combinations. First we get one team out of 8 . The number of ways to do this would be \(C_8^2\). The next combination is 2 out of 6 or \(C_6^2\), and so on. Having all four combinations multiplied, we need to divide the total number by the number of ways the teams can be chosen \(4!\), since we are not interested if the team with two certain people is chosen first, second or third. Therefore, the answer is found by the following formula \(\frac{C_8^2*C_6^2*C_4^2*C_2^2}{4!}=105\).
Answer: B
Bunuel, can you please explain why the total no. of ways is divided by 4! when we are not interested if the team with two certain people is chosen first, second or third? I mean how have you determined that it should be divided by 4! ?
Another way to think about this is remove the: - meaning for order within the groups - meaning for order among the groups solution: 1. we arrange all the 8 people in a line, this gives us: 8! 2. since the order among the gourps does not matter, we will have to devide 8! by 4! 3. since the order within the groups allso does not matter, for each group, we will have to devide by 2!
hence we got: 8!/(4!2!2!2!2!) = 105
comment: by saying "the order does not matter" i mean that, for example if we take the order within the gourp, it does matter if i picked person X to be the first in the group and later person Y to be in the group.
To combine the two approaches. I guess by solving to get 2520, we are doing stage 1 and 3 of your approach. Dividing by 4! is the stage 2...
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