As Part Of a Game, Four People Each Must Secretly Choose an Integer Between 1 and 4 GMAT Problem Solving

Question: As part of a game, four people each must secretly choose an integer between 1 and 4, inclusive. What is the approximate likelihood that all four people will choose different numbers?

1. 9%
2. 12%
3. 16%
4. 20%
5. 25%

‘As part of a game, four people each must secretly choose an integer between 1 and 4, inclusive’ - is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

Solution and Explanation:

Approach Solution 1:

Given:

• Four people each must secretly choose an integer between 1 and 4, inclusive.

Find Out:

• Approximate likelihood that all four people will choose different numbers

The first guy can choose anything
Hence, the chances of success is 1/1
Considering that the first guy already choose a number, the second guy can choose any three of the four numbers.
So the chance of success becomes 3/4
Similarly, the third guy can choose any two of the four numbers
The chance of success are 2/4
The last remaining guy can choose only one of the four numbers
chance of success 1/4
The Probability is :
1 x 3/4 x 2/4 x 1/4
= 6/64
=3/32
=9.37%
The nearest option is 9% and hence, A is the correct answer.

Correct Answer: A

Approach Solution 2:
This is the second way via which the problem can be solved.
Given:

• Four people each must secretly choose an integer between 1 and 4, inclusive.

Find Out:

• Approximate likelihood that all four people will choose different numbers

Let us consider that the first person chooses any of 4 numbers.
The second person can choose one number from the left over 3.
The third can choose one number from the left over two
The last person chooses the last remaining number.
Hence, there are multiple ways in which the number can be chosen. The ways are:
=>4 * 3 * 2 * 1 = 24 ways
Since there are 4 numbers, total possible chances are:
4 * 4 * 4 * 4 = 256
Thus probability is = (24/256) * 100 = 9%
Hence, A is the correct answer.

Correct Answer: A

Approach Solution 3

It is given that 4 people have the option of choosing 1,2,3 or 4. So, the 1st person can choose any number and the probability of that is 1. Then the 2nd person can only choose 3 of the 4 numbers. Therefore the probability of that is ¾. The third person can only choose 2 of the 4 numbers and the probability of that is 2/4. Hence, the final person can only choose 1 of the 4 numbers.  The probability of that is ¼. 
So,
1 * (3/4) * (2/4) * (1/4) = 3/32
~9%

Correct Answer: A

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