Data Sufficiency

ardz24 wrote:Time for a brand-new quant problem, ladies and gentlemen!

A firm has at least 5 partners, of whom at least 2 are male and at least 2 are female. If four of this firm's partners are to be selected at random for an audit, what is the probability that equal numbers of male and female partners will be selected?

(1) The firm has no more than 3 male partners.
(2) The firm has no more than 3 female partners.

What's the best way to determine which statement is sufficient? Can any experts help?

We have the number of partners ≥ 5.

(1) The firm has no more than 3 male partners.

We do not have any information about the number of partners. Since as per the statement, we have 2 or 3 males, but the number of females can be any number. This way, we cannot determine the required answer. Insufficient.

(2) The firm has no more than 3 female partners.

As with Statement 1, this statement is also not sufficient.

(1) and (2) combined:

From Statement 1, we have the number of males = 2 or 3 and from Statement 2, we have the number of females = 2 or 3.

Since the total number of partners ≥ 5, if the number of males = 2, the number of females = 3 and vice-versa.

Case 1: Say, the number of males = 2 and the number of females = 3, thus, the total number of partners = 5

Probability = (2C2*3C2) / 5C4 = (1*3C1) / 5C1 = 3/5; we know that nCr = nC(n-r)

Case 1: Say, the number of females = 2 and the number of males = 3, thus, the total number of partners = 5

Probability = (2C2*3C2) / 5C4 = (1*3C1) / 5C1 = 3/5. Same answser. Sufficient.

The correct answer: C

Hope this helps!

-Jay
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