Question: A dog breeder currently has 9 breeding dogs. 6 of the dogs have exactly 1 littermate, and 3 of the dogs have exactly 2 littermates. If 2 dogs are selected at random, what is the probability that both selected dogs are NOT littermates?

1. 1/6
2. 2/9
3. 5/6
4. 7/9
5. 8/9

Correct Answer: C

Solution and Explanation:
Approach Solution 1:
Since there are 6 dogs with exactly 1 littermate each, then we have three pairs of littermates: (1-2), (3-4), (5-6);
Since there are 3 dogs with exactly 2 littermates each, then we have one triple of littermates: (7-8-9).

Let's find the probability of the opposite event, so the probability that both selected dogs ARE littermates, and subtract it from 1.

We can select either (1-2), (3-4), or (5-6) so 3 ways;
We can select any two from (7-8-9) so C^3/2=3 ways;
So, total ways to select 2 littermates out of these 9 dogs is 3+3=6;

Total # of ways to select 2 dogs out of 9 is C^9/2=36;
P=1-6/36=5/6.

Approach Solution 2:
Since 6 have exactly 1 littermate ,
What we really have are 3 relationships amongst 6 .
1->2
3->4
5->6

Since 3 have exactly 2 relationships.
What we really have are 3 relationships.
7->8
8->9
7->9

Total of 6 relationships.
Probability they will be related = number of relationships/ total number of ways u can pick 2 from 9
= 6/ 9C2
=6/36=1/6

Hence Probability that they are not related is = 1- 1/6 = 5/6

Approach Solution 3:
1) select no littermates from Group 1 (1-2), (3-4), (5-6)
6*2=12 (i.e. 1-3; 1-4; 1-5;1-6;2-3;2-4;2-5;2-6;3-5;3-6;4-5;4-6)
2) select no littermates from Group 2 (7-8-9) . it equals to zero
3) select a dog from Group 1 and one from Group 2 (again no littermate)
3*6=18
(12+0+18)/9C2=30/36=5/6

“A dog breeder currently has 9 breeding dogs. 6 of the dogs have”- is a topic of the GMAT Quantitative reasoning section of GMAT. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. GMAT Quant practice papers improve the mathematical knowledge of the candidates as it represents multiple sorts of quantitative problems.

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