Combinations and Probability

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April 4th 2010, 09:59 PM
pozat

Combinations and Probability

Question

A panel of 5 members is drawn from a group consisting of 2 managers, 7 assistants and 4 secretaries.

(a) How many ways can the panel be formed with at least 1 manager and 2 assistants

(b) Find the probability that the panel include 2 managers, 2 assistants and 1 secretary

(Bow)

Answer

(a) Should I use the following approach?

1M+2A+2S: C(2,1)*C(7,2)*C(4,2)=252
1M+3A+1S: C(2,1)*C(7,3)*C(4,1)=280
1M+4A+0S: C(2,1)*C(7,4)=70
2M+2A+1S: C(2,2)*C(7,2)*C(4,1)=84
2M+3A+0S: C(2,2)*C(7,3)=35

By addition, there are 721 combinations.

(b) P=2/13*1/12*7/11*6/10*4/9=14/6435
April 4th 2010, 10:02 PM
Anonymous1

Rather than working out all the cases, simply choose the other 2 committee members from the remaining group of 10 people.

$a)$ ${2 \choose 1}{7 \choose 2}{10 \choose 2}$

$b)$ $\frac{{2 \choose 2}{7 \choose 2}{4 \choose 1}}{{13\choose 5}}$
April 4th 2010, 10:42 PM
matheagle

It's.... at least 1 manager and 2 assistants
but I'm not sure if that means at least 1 manager and exactly 2 assistants
or at least 1 manager and at least 2 assistants.

Prosac's (a) is ok
In (b) you're putting in order and you need to eliminate that.

Annoying-mouse's (b) is ok
You didn't miss the exactly, but I don't think it's correct either.
April 4th 2010, 10:57 PM
Anonymous1

Quote:

Originally Posted by matheagle

It's.... at least 1 manager and 2 assistants
but I'm not sure if that means at least 1 manager and exactly 2 assistants
or at least 1 manager and at least 2 assistants.

Prosac's (a) is ok
In (b) you're putting in order and you need to eliminate that.

Annoying-mouse's (b) is ok
In (a) you missed the at least, you did exactly 1 ,2, 2.

Your so dreamy haha.
April 4th 2010, 11:02 PM
Anonymous1

Quote:

Originally Posted by matheagle

In (a) you missed the at least, you did exactly 1 ,2, 2.

Can't the ${10\choose 2}$ be of type manager, assistant or secretary though? And, therein, un-fixing the exactly-ness?
April 4th 2010, 11:28 PM
pozat

Double confirmed with my teacher and classmates, it should be asking at least 1 manager and at least 2 assistants.

I got what Ma-the-agleam means. I wrongly put (b) in order. Annoying-mouse's answer is GREAT.

Thanks to ALL ~

(Clapping)