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Alice and David are two of the ten people in a group. In how many diff

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Alice and David are two of the ten people in a group. In how many diff  [#permalink]

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New post 14 Aug 2018, 23:51
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Alice and David are two of the ten people in a group. In how many different ways can this group of 10 people be divided into a group of 7 people and a group of 3 people if Alice and David are to be in the same group?


A. 48

B. 56

C. 64

D. 80

E. 128

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Re: Alice and David are two of the ten people in a group. In how many diff  [#permalink]

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New post 15 Aug 2018, 01:04
Bunuel wrote:
Alice and David are two of the ten people in a group. In how many different ways can this group of 10 people be divided into a group of 7 people and a group of 3 people if Alice and David are to be in the same group?


A. 48

B. 56

C. 64

D. 80

E. 128


There are two scenarios

Case 1: When Alice and David are in group of 7 : Total ways to make such groups of 7 and 3 = 8C5 (To select 5 more out of 8 to make group of 7 which includes Alice and David already)

This way the other group of 3 is automatically formed, 8C5 = 56

Case 2: When Alice and David are in group of 3 : Total ways to make such groups of 3 and 7 = 8C1 (To select 1 more out of 8 to make group of 3 which includes Alice and David already)

This way the other group of 7 is automatically formed, 8C1 = 8

Total Possible ways = 56+8 = 64

Answer: Option C
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Re: Alice and David are two of the ten people in a group. In how many diff  [#permalink]

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New post 15 Aug 2018, 01:17
64

1*1*8c5+1*1*8c1

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Re: Alice and David are two of the ten people in a group. In how many diff  [#permalink]

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New post 16 Aug 2018, 01:14
1

Solution


Given:
    • There are 10 people in a group
    • Alice and David are two among the ten people

To find:
    • In how many ways can these 10 people be divided into a group of 7 and a group of 3, such that Alice and David are in the same group

Approach and Working:
There are two possible scenarios,
    • Scenario 1: Alice and David are in the group of 7 people
    • Scenario 2: Alice and David are in the group of 3 people

In Scenario 1, the total number of ways will be \(^8C_5\)
    • Since, Alice and David are two among the 7 people, the other 5 people can be selected from the remaining 8 people in 8C5 ways.
    • Later the remaining 3 will form another group.

In Scenario 2, the total number of ways will be \(^8C_1\)
    • Since, Alice and David are two among the 3 people, the other person can be selected from the remaining 8 people in 8C1 ways.
    • Later the remaining 7 will form another group.

Therefore, the total number of ways = \(^8C_5 + ^8C_1 = 8 + 56 = 64\)

Hence, the correct answer is option C.

Answer: C

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Re: Alice and David are two of the ten people in a group. In how many diff  [#permalink]

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New post 16 Aug 2018, 01:38
OA:C

No of ways, when Alice and David are in different groups \(: 2 * C(8,6) * C(2,2) = 2 *\frac{8!}{6!*2!}*1=56\)

Total number to form two teams one of \(7\) members and another \(3\) members, out of total \(10\) members, without any restrictions

\(= C(10,7)*C(3,3)=\frac{10!}{7!*3!}*1=120\)

Numbers of ways when Alice and David are in the same team = Total number of ways, without restrictions - No of cases when they are in different teams

\(=120-56 = 64\)
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Re: Alice and David are two of the ten people in a group. In how many diff &nbs [#permalink] 16 Aug 2018, 01:38
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