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# There are 10 people working at Violet Lazer Inc. Among them

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SVP
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There are 10 people working at Violet Lazer Inc. Among them [#permalink]

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06 Nov 2003, 06:03
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There are 10 people working at Violet Lazer Inc. Among them are President and two VPs. A three-person committee is to be formed from all the people provided that:

(1) President do not go with either VP;
(2) Two VPs cannot go together.

In how many ways can the committee be formed?
SVP
Joined: 03 Feb 2003
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06 Nov 2003, 06:33
the two conditions are simultaneously applied
CEO
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23 Nov 2003, 22:31
stolyar wrote:
no objection

stolyar

I get 96.

could you explain how you get 98.

thanks
Intern
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24 Nov 2003, 23:45
I also got 96.

Total ways = 10C6 = 120

Ways in which President is with vp = 1 (president) * 2(VP) * 8(others) = 16

Ways in which both VPs are together = 1 (vp) * 1 (vp) * 8 (others) = 8

So, answer = 120 - 16 - 8 = 96.
Director
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25 Nov 2003, 03:31
when you select Pr. you need to select 2 out of 7 which is 21 ways.
you can select any of 2 VPr. in 2 ways and 2 out of 7 in 21 or total 42 ways.
you can select 3 out of 7 ( no Vp,no Pr.) in 35 ways or total 21+42+35=98 ways
Manager
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26 Nov 2003, 11:22
sujayath wrote:
I also got 96.

Total ways = 10C6 = 120

Ways in which President is with vp = 1 (president) * 2(VP) * 8(others) = 16

Ways in which both VPs are together = 1 (vp) * 1 (vp) * 8 (others) = 8

So, answer = 120 - 16 - 8 = 96.

why is total ways 10C6, aren't we selecting 3 from 10 thus 10C3???please explain
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26 Nov 2003, 13:47
I go with BG: 98

I cant understand the logic of 96.

Here we go:

We need to select 3 ppl.
Option 1:
Choose 3 from 7 (No P or VP) - 7C3 = 35
Option 2:
P fills one of the positions - then VPs are out of question, hence 7C2 ways of choosing remaining 2 ppl from the 7. = 1C1 * 7C2 = 21
Option 3:
VPs fill one position, 2C1 & 7C2 fills the remaining 2 slots.
= 2C1 * 7C2 = 42

Hence 35+21+42 = 98.

Now coming to the 96 solution: It should be 98 too using the same explaination provided.
10C3 = 120 choose 3 out of 10
But the following cannot happen:
1. P + 2Vps = 1C1 * 2C2 = 1
2. P + 1 VP + 1 Other = 1C1 * 2C1 * 7C1 = 1 * 2 * 7 = 14
3. 2VPs + 1 Other = 2C2 * 7C1 = 1 * 7 = 7

So the total of above is 22

And 120 -22 = 98

The logic being used for 96 is:
Ways in which President is with vp = 1 (president) * 2(VP) * 8(others) = 16
Ways in which both VPs are together = 1 (vp) * 1 (vp) * 8 (others) = 8

8 is not a number to be considered at all. In any case the choice will be made between 1 P, 2 VP, 7 Others.

Cheers
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04 Dec 2003, 06:41
the total numer of combinations is 10C3=120
the wrong combinations are P+VP+one worker and P+VP+VP and VP+VP+one worker
[1*2*7]+[1*1*1]+[1*1*7]=14+1+7=22

120-22=98
04 Dec 2003, 06:41
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