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A certain law firm consists of 4 senior partners and 6 [#permalink]

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19 Nov 2007, 06:21

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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)

Re: A certain law firm consists of 4 senior partners and 6 [#permalink]

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17 Jul 2013, 14:49

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Maxirosario2012 wrote:

A certain law firm consists of 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. (Two groups are considered different if at least one group member is different)

48 100 120 288 600

Another approach:

SSSS JJJJJJ

We have 4 seniors and 6 juniors. We are asked for the nomber of groups of 3 in which at least 1 is a senior (1, 2 or 3 seniors in each group). Considering that we have 3 types of groups:

Groups with 1 senior: We take 1 senior out of 4 (\(C^4_1\)) and combine them with 2 juniors out of 6 (\(C^6_2\)):

\(C^4_1*C^6_2 = 4*15 = 60\)

Groups with 2 seniors: We take 2 seniors out of4 (\(C^4_2\)) and combine them with 1 juniors out of 6 (\(C^6_1\)):

\(C^4_2*C^6_1 = 6*6 = 36\)

Groups with 3 seniors (and zero juniors): We take 3 seniors out of 4 (\(C^4_3\)):

\(C^4_3\) = 4

60 + 36 + 4 = 100

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

Total # of different groups of 3 out of 10 people: \(C^3_{10}=120\); # of groups with only junior partners (so with zero senior member): \(C^3_6=20\);

So the # of groups with at least one senior partner is {all} - {none}= {at least one} = 120-20 = 100.

Total no of groups of 3 members (including junior and senior) = 10C3
Total no of groups of 3 members (only juniors) = 6C3
Total no of groups of 3 members (at least 1 senior) = 10C3 - 6C3 = 120 - 20 =100

A certain law firm consists of 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. (Two groups are considered different if at least one group member is different)

48 100 120 288 600

My take :

4C1 * 6C2 + 4C2 *6C1

pLEASE ADVICE. tHANKS

im really lost on this problem...

What really helps me in these type of problems is to approach the thought process like this:

1. What are the total # of combinations without any restrictions ?

2. What are the total # of combinations using the OPPOSITE of the restriction ?

3. The # of combinations with the restriction is the difference between 1 and 2

So, for this question, 1 would be 10C3. 2 would be 6C3, which gives the # of ways to pick 3 ppl out of the 6 junior partners, i.e. no seniors.

Re: A certain law firm consists of 4 senior partners and 6 [#permalink]

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17 Jul 2013, 13:49

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A certain law firm consists of 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. (Two groups are considered different if at least one group member is different)

48 100 120 288 600

Another approach:

SSSS JJJJJJ

We have 4 seniors and 6 juniors. We are asked for the nomber of groups of 3 in which at least 1 is a senior (1, 2 or 3 seniors in each group). Considering that we have 3 types of groups:

Groups with 1 senior: We take 1 senior out of 4 (\(C^4_1\)) and combine them with 2 juniors out of 6 (\(C^6_2\)):

\(C^4_1*C^6_2 = 4*15 = 60\)

Groups with 2 seniors: We take 2 seniors out of4 (\(C^4_2\)) and combine them with 1 juniors out of 6 (\(C^6_1\)):

\(C^4_2*C^6_1 = 6*6 = 36\)

Groups with 3 seniors (and zero juniors): We take 3 seniors out of 4 (\(C^4_3\)):

\(C^4_3\) = 4

60 + 36 + 4 = 100 _________________

Encourage cooperation! If this post was very useful, kudos are welcome "It is our attitude at the beginning of a difficult task which, more than anything else, will affect It's successful outcome" William James

A certain law firm consists of 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. (Two groups are considered different if at least one group member is different)

A certain law firm consists of 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. (Two groups are considered different if at least one group member is different)

48 100 120 288 600

My take :

4C1 * 6C2 + 4C2 *6C1

pLEASE ADVICE. tHANKS

Your method is correct but you have not considered the possibility of all the 3 members of the group being senior members.

A certain law firm consists of 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. (Two groups are considered different if at least one group member is different)

48 100 120 288 600

My take :

4C1 * 6C2 + 4C2 *6C1

pLEASE ADVICE. tHANKS

im really lost on this problem...

What really helps me in these type of problems is to approach the thought process like this:

1. What are the total # of combinations without any restrictions ?

2. What are the total # of combinations using the OPPOSITE of the restriction ?

3. The # of combinations with the restriction is the difference between 1 and 2

So, for this question, 1 would be 10C3. 2 would be 6C3, which gives the # of ways to pick 3 ppl out of the 6 junior partners, i.e. no seniors.

The final answer should be 10C3 - 6C3

pmenon great explanation!! I totally lost this one.

A certain law firm consists of 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. (Two groups are considered different if at least one group member is different)

A certain law firm consists of 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. (Two groups are considered different if at least one group member is different)

48 100 120 288 600

My take :

4C1 * 6C2 + 4C2 *6C1

pLEASE ADVICE. tHANKS

Ans = Total Comb - Comb with only junior partners = 10c3 - 6c3 = 100 (B) _________________

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Re: PS - Combination
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16 Feb 2010, 13:32

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