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# A firm has 4 senior partners and 6 junior partners. How many

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Re: A firm has 4 senior partners and 6 junior partners. How many  [#permalink]

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22 May 2018, 11:12
Hi Manychips,

In your calculation, you assume that the first person chosen MUST be a Senior Partner, but then you divide the entire calculation by 3! (which is something that you can only do if the first person can be ANY partner - Senior or Junior). If you want to approach the question in this way, then you would have to do the following:

Total number of PERMUTATIONS (regardless of whether the member is Senior or Junior) = (10)(9)(8) = 720
Total number of UNIQUE groups of three = 720/3! = 120

Total Permutations of three that are ONLY Junior Members = (6)(5)(4) = 120
Total number of UNIQUE groups of Junior Members = 120/3! = 20

Total groups with AT LEAST 1 Senior Member = (Total of All Groups) - (Total of JUST Junior Members) = 120 - 20 = 100

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Re: A firm has 4 senior partners and 6 junior partners. How many  [#permalink]

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28 May 2018, 21:22
Please tell me what I have done wrong
First , we choose a Senior member :
C(4,1) =4
then we choose 2 members from the rest (9)
C(9,2) =36
so the number of methods is 4*36 = 144

Wrong but why ?
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Re: A firm has 4 senior partners and 6 junior partners. How many  [#permalink]

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28 May 2018, 22:11

Unfortunately, your math includes some 'duplicate entries.'

For example, let's call the 4 senior partners A, B, C and D and the 6 junior partners 1, 2, 3, 4, 5 and 6.

In your calculation, you state that the first person selected MUST be one of those 4 seniors (A/B/C/D) and the remaining two people can be any two of the remaining 9...

The group "A/B/1" and "B/A/1" are the SAME group, but your calculation counts THAT group TWICE (depending on whether A or B was chosen first). In a Combination question, you can't allow duplicate entries.

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Re: A firm has 4 senior partners and 6 junior partners. How many  [#permalink]

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20 Aug 2019, 05:07
gdk800 wrote:
A firm has 4 senior partners and 6 junior partners. How many different groups of 3 partners can be formed in which at least one member of the group is a senior partner. (2 groups are considered different if at least one group member is different)

A. 48
B. 100
C. 120
D. 288
E. 600

Bunuel VeritasKarishma I got a doubt regarding an alternate approach. Would really appreciate your help

Let Senior Partner = S, Junior Partner = J

Approach: Choose 1 S and then consider the other cases.

Ways of choosing 1 S: 4C1 = 4
Then we have 3 S left

Now the possible cases of picking 2 people from 3S and 6J are: 1) 1S + 1J 2) 0S + 2J 3) 2S + 0J
Total # for those 3 cases = 3C1 * 6C1 + 6C2 + 3C2 = 3*6 + 15 + 3 = 36

Total cases = 4 * 36 = 144

What am I doing wrong since the answer doesn't match the OA? Thanks!
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A firm has 4 senior partners and 6 junior partners. How many  [#permalink]

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20 Aug 2019, 06:31
gdk800 wrote:
A firm has 4 senior partners and 6 junior partners. How many different groups of 3 partners can be formed in which at least one member of the group is a senior partner. (2 groups are considered different if at least one group member is different)

A. 48
B. 100
C. 120
D. 288
E. 600

Given: A firm has 4 senior partners and 6 junior partners.

Asked: How many different groups of 3 partners can be formed in which at least one member of the group is a senior partner?

Number of groups in which at least one member is a senior partner = Total number of groups - Number of groups in which no member is a senior partner

Total number of groups $$= ^{10}C_3 = 120$$
Number of groups in which no member is a senior partner $$= ^6C_3 = 20$$

Number of groups in which at least one member is a senior partner = Total number of groups - Number of groups in which no member is a senior partner
= 120 - 20 = 100

IMO B
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Re: A firm has 4 senior partners and 6 junior partners. How many  [#permalink]

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21 Aug 2019, 01:13
1
dabaobao wrote:
gdk800 wrote:
A firm has 4 senior partners and 6 junior partners. How many different groups of 3 partners can be formed in which at least one member of the group is a senior partner. (2 groups are considered different if at least one group member is different)

A. 48
B. 100
C. 120
D. 288
E. 600

Bunuel VeritasKarishma I got a doubt regarding an alternate approach. Would really appreciate your help

Let Senior Partner = S, Junior Partner = J

Approach: Choose 1 S and then consider the other cases.

Ways of choosing 1 S: 4C1 = 4
Then we have 3 S left

Now the possible cases of picking 2 people from 3S and 6J are: 1) 1S + 1J 2) 0S + 2J 3) 2S + 0J
Total # for those 3 cases = 3C1 * 6C1 + 6C2 + 3C2 = 3*6 + 15 + 3 = 36

Total cases = 4 * 36 = 144

What am I doing wrong since the answer doesn't match the OA? Thanks!

You are double counting cases. Whenever you make partial selections from a group for another group, you double count. Here, you first selected an S for your team and then selected 1 S or 2S or 0S again from the group of S for your team.

So say the 4 S are Sa, Sb, Sc and Sd. You select Sa.
Now in your first case, you have to select another S. Say you select Sc. Then you select Ja from juniors.

Now think of another case. First you select Sc.
Now in your first case, you select Sa. Then you select Ja from juniors.

Both these teams are the same but you counted them as 2. That's your error.
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Re: A firm has 4 senior partners and 6 junior partners. How many  [#permalink]

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21 Aug 2019, 02:13
gdk800 wrote:
A firm has 4 senior partners and 6 junior partners. How many different groups of 3 partners can be formed in which at least one member of the group is a senior partner. (2 groups are considered different if at least one group member is different)

A. 48
B. 100
C. 120
D. 288
E. 600

3 people can be selected out of 10people in 10C3 ways =120 ways.

Out of these 120 ways there will be some group where there is no senior.

No of groups where there is no senior = 6C3= 20

So groups where there will be at least one senior = 120 -20
=100

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Re: A firm has 4 senior partners and 6 junior partners. How many  [#permalink]

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23 Aug 2019, 08:09
gdk800 wrote:
A firm has 4 senior partners and 6 junior partners. How many different groups of 3 partners can be formed in which at least one member of the group is a senior partner. (2 groups are considered different if at least one group member is different)

A. 48
B. 100
C. 120
D. 288
E. 600

we get 3 possiblities ; 4c1*6c2+ 4c2*6c1+ 4c3 ; 100
IMO B
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Re: A firm has 4 senior partners and 6 junior partners. How many   [#permalink] 23 Aug 2019, 08:09

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