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# A company has 3 directors and 5 managers. How many different

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Intern
Joined: 15 Jun 2010
Posts: 9
A company has 3 directors and 5 managers. How many different  [#permalink]

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Updated on: 16 Apr 2014, 23:55
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Difficulty:

35% (medium)

Question Stats:

67% (02:03) correct 33% (02:44) wrong based on 73 sessions

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A company has 3 directors and 5 managers. How many different committees with 5 people can be chosen having at least 1 director?

A. 500
B. 720
C. 4500
D. 25
E. 55

Originally posted by tauchmeister on 16 Apr 2014, 14:53.
Last edited by Bunuel on 16 Apr 2014, 23:55, edited 1 time in total.
Edited the question
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Re: A company has 3 directors and 5 managers  [#permalink]

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16 Apr 2014, 18:33
3
atleast one means exactly one and more than one.

different committees with 5 people can be chosen having at least 1 director is
(3c1*5c4) + (3c2*5c3) + (3c3*5c2) = 55;
Hence E.
Press Kudos if this helps
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Joined: 02 Sep 2009
Posts: 64939
Re: A company has 3 directors and 5 managers. How many different  [#permalink]

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16 Apr 2014, 23:59
1
tauchmeister wrote:
A company has 3 directors and 5 managers. How many different committees with 5 people can be chosen having at least 1 director?

A. 500
B. 720
C. 4500
D. 25
E. 55

The number of committees with at least 1 director = (total number of committees) - (number of committees with 0 directors) = $$C^5_8-C^5_5=56-1=55$$.

Hope it's clear.
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Re: A company has 3 directors and 5 managers. How many different  [#permalink]

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21 Jun 2019, 10:35
tauchmeister wrote:
A company has 3 directors and 5 managers. How many different committees with 5 people can be chosen having at least 1 director?

A. 500
B. 720
C. 4500
D. 25
E. 55

We can use the equation:

Number of committees with at least director = total number of committees - number of committees with no directors.

The total number of committees can be formed in 8C5 ways:

8C5 = 8!/(5! x 3!) = (8 x 7 x 6 x 5 x 4)/(5 x 4 x 3 x 2) = 8 x 7 = 56
The number of committees that can be formed with no directors is 5C5 = 1.

Thus, the number of committees with at least one director is 56 - 1 = 55.

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Re: A company has 3 directors and 5 managers. How many different   [#permalink] 21 Jun 2019, 10:35