## Triplets Adam, Bruce, and Charlie enter a triathlon. If there are 9 competitors in the triathlon, and medals are awarded

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### Triplets Adam, Bruce, and Charlie enter a triathlon. If there are 9 competitors in the triathlon, and medals are awarded

by BTGmoderatorDC » Wed Oct 20, 2021 6:19 pm

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Triplets Adam, Bruce, and Charlie enter a triathlon. If there are 9 competitors in the triathlon, and medals are awarded for first, second, and third place, what is the probability that at least two of the triplets will win a medal?

A. 3/14
B. 19/84
C. 11/42
D. 15/28
E. 3/4

OA B

Source: Manhattan Prep

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### Re: Triplets Adam, Bruce, and Charlie enter a triathlon. If there are 9 competitors in the triathlon, and medals are awa

by [email protected] » Thu Oct 21, 2021 9:37 am
BTGmoderatorDC wrote:
Wed Oct 20, 2021 6:19 pm
Triplets Adam, Bruce, and Charlie enter a triathlon. If there are 9 competitors in the triathlon, and medals are awarded for first, second, and third place, what is the probability that at least two of the triplets will win a medal?

A. 3/14
B. 19/84
C. 11/42
D. 15/28
E. 3/4

OA B

Source: Manhattan Prep
P(at least two triplets win a metal) = P(exactly two triplets win a middle OR exactly three triplets win a medal)
= P(exactly two win a middle) + P(exactly three win a medal)

Let's examine each probability separately....

P(exactly two triplets win a middle)
There are three ways for this to happen:
case i) a triplet places 1st, another triplet places 2nd, a non-triplet places 3rd
case ii) a triplet places 1st, a non-triplet places 2nd, a triplet places 3rd
case iii) a non-triplet places 1st, a triplet places 2nd, a non-triplet places 3rd

P(case i) = (3/9)(2/8)(6/7) = 1/14
P(case ii) = (3/9)(6/8)(2/7) = 1/14
P(case iii) = (6/9)(3/8)(2/7) = 1/14
P(exactly two triplets win a middle) = 1/14 + 1/14 + 1/14 = 3/14

P(exactly three triplets win a middle)
P(exactly three triplets win a middle) = P(a triplet places 1st, another triplet places 2nd, another triplet places 3rd)
= (3/9)(2/8)(1/7) = 1/84

We get:
P(at least two triplets win a metal) = P(exactly two win a middle) + P(exactly three win a medal)
= 3/14 + 1/84
= 18/84 + 1/84
= 19/84