Question: In a drawer, there are 4 white socks, 3 blue socks, and 5 grey socks. Two socks are picked randomly. What is the possibility that both the socks are of the same colour?
- 1
- \(\frac{17}{21}\)
- \(\frac{13}{17}\)
- \(\frac{19}{66}\)
- \(\frac{1}{2}\)
“In a drawer, there are 4 white socks, 3 blue socks, and 5 grey socks” is the topic from the GMAT Problem Solving question. GMAT quantitative reasoning section tests the candidate's ability to solve mathematical problems, interpret graph data, and mathematical reasoning. The GMAT quantitative exam consists of 31 questions. In a drawer there are 4 white socks, 3 blue socks and 5 grey socks has five options to answer from. This GMAT Problem-solving question has five options to select from. In this section candidates get MCQs having five options out of which only one is true.
Solution and Explanation:
Approach Solution 1:
Given to us that there are 4 white socks, 3 blue socks, and 5 grey socks.It is said that two socks are picked randomly. It is asked to find out the probability that both socks are of the same colour.
This is a question from the topic: permutation and combination and probability.
In order to solve the problem, the candidate should remember the following formulas for permutation and combination.
Permutation of n objects taken r at a time: \(^nP_r\)= n! / (n-r)!
Combination of n objects taken r at a time: \(^nC_r\) = n! / ( (n-r)! * r!)
where x! = x*(x-1)*(x-2)*......(3).(2).(1)
Here firstly it is important to find out how many ways there are to pick up two socks from all the socks present.
Total number of socks = no of white socks + no of blue socks + number of grey socks
= 3 + 4 + 5
= 12 socks
There are a total of 12 socks.
No of ways to select two socks from these 12 socks =
\(^{12}C_2\)= 12! / ((12-2)! 2!) = 12! / (10! * 2!) = (12 * 11 * 10!) / (10! * 2)
= 12 * 11/2 = 66
There are 66 ways to select 2 socks from a set of 12 socks.
Now we have to find the number of ways to select two socks of the same colour.
So either we can choose two socks of white colour or 2 socks of blue colour or 2 socks of grey colour.
No of ways to choose 2 socks of same colour = \(^4C_2+{^3C_2}+{^5C_2}\)
= 4! / (2!*2!) + 3!/(1!* 2!) + 5!/(3! * 2!)
= 12/2 + 3 + (5*4)/2
= 6 + 3 + 10
= 19
The probability of choosing 2 socks of same colour from a given set of 4 white socks, 3 blue socks, and 5 grey socks is - 19/66
So option D is the correct answer.
Correct Answer: D
Approach Solution 2:
It is a method of elimination to find out the answer.
It should be noted that whichever colour you pick first, less than half of the remaining socks will be of the matching colour.
So this implies that the answer must be less than half.
There is only one option satisfying the given condition which is option D.
So D is the correct answer.
Method of elimination helps to solve problems quickly in less time by eliminating the options.
Correct Answer: D
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