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Six couples are invited to play a game of cards, which needs 4 players

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Six couples are invited to play a game of cards, which needs 4 players  [#permalink]

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New post Updated on: 06 Jun 2020, 03:00
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Six couples are invited to play a game of cards, which needs 4 players at a time. In how many ways can the players be selected, if no couple should be included?

(A) 256
(B) 384
(C) 240
(D) 320
(E) 60

Originally posted by Keyurneema on 06 Jun 2020, 02:55.
Last edited by Bunuel on 06 Jun 2020, 03:00, edited 1 time in total.
Renamed the topic.
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Six couples are invited to play a game of cards, which needs 4 players  [#permalink]

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New post Updated on: 11 Jun 2020, 19:52
Six couples but no couple can play card game at a time. i.e. only a single person form each couple pair
Also, only 4 players can play the game at a time.

Step 1
We start with selecting 4 couples out of 6 couples. (Because only 4 players can play at one time.)

This is done in \(6C4\) ways. ===> \(6C4 = 6!/(4!*2!) \) = 15 ways.

Step 2
Now that we have selected the 4 couples, we select one from the pair who actually plays the card game.
(Because no couple should be included i.e. only one person from a couple pair can be selected)

4 couples, we can select 1 person from each pair. i.e. 2 choices per couple.
This is done is \(2^4\) = \(2*2*2*2\) = 16 ways.

Both conditions need to be satisfied, so it's an AND case of combination.
Total ways = 15*16 = 240 ways.

Answer C.
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Originally posted by Rajat8 on 06 Jun 2020, 06:26.
Last edited by Rajat8 on 11 Jun 2020, 19:52, edited 1 time in total.
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Re: Six couples are invited to play a game of cards, which needs 4 players  [#permalink]

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New post 06 Jun 2020, 07:07
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Keyurneema wrote:
Six couples are invited to play a game of cards, which needs 4 players at a time. In how many ways can the players be selected, if no couple should be included?

(A) 256
(B) 384
(C) 240
(D) 320
(E) 60

Total cases= 12C4=495
Cases which are not required= 2 couples or 1 couple and 2 others(not couple)
2 couples= 6C2=15
1 couple + 2 others=6C1*(1*8C1*5)=240 ( after selecting one couple among 6, we are left with 5 couples. Now select anyone( 1 way), then we are left with 8 people as we can not select the his or her partner. And we can select 1 person among 8 people in 8C1 ways and we have 5 such couples. So 1*8C1*5)
Cases which are not required=255
Required cases=495-255
=240
C:)
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Re: Six couples are invited to play a game of cards, which needs 4 players  [#permalink]

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New post 06 Jun 2020, 08:12
Answer: Option C


No couples allowed at one time, so one player from a couple can be selected at one time

\(=> 6C4 =\frac{ 6!}{(4! * 2!)}
=> 15 ways\)

Each player from a couple can be selected in 2 ways => either first or second and there are 4 couples so, \( 2 ^ 4 = 16 ways\)

Total ways = \(15 * 16 = 240 ways\) (C)
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Re: Six couples are invited to play a game of cards, which needs 4 players  [#permalink]

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New post 06 Jun 2020, 08:16
* Select 4 couple out of 6 couple- 6C4, & Than selecting 1 person from a couple because both can not Play together- 2 WAYS

=6C4*2*2*2*2= 15*16= 240

Ans -C
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Re: Six couples are invited to play a game of cards, which needs 4 players  [#permalink]

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New post 06 Jun 2020, 08:29
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Imagine first we're picking a first player, second player, third player and fourth player. We'd have 12 choices for the first player, 10 for the second (to avoid picking a couple), 8 for the third and 6 for the fourth. So we'd have (12)(10)(8)(6) choices. But presumably the order of the 4 players is irrelevant, so we must divide by 4!, and we get

(12)(10)(8)(6)/(4!) = (5)(8)(6) = 240 possible selections
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Re: Six couples are invited to play a game of cards, which needs 4 players  [#permalink]

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New post 06 Jun 2020, 09:44
Keyurneema wrote:
Six couples are invited to play a game of cards, which needs 4 players at a time. In how many ways can the players be selected, if no couple should be included?

(A) 256
(B) 384
(C) 240
(D) 320
(E) 60


method 1
12*10*8*6/4! = 240
method 2
6c3*2*2*2*2 ; 15*16 ; 240
OPTION C
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Re: Six couples are invited to play a game of cards, which needs 4 players   [#permalink] 06 Jun 2020, 09:44

Six couples are invited to play a game of cards, which needs 4 players

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