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The only contents of a container are 10 disks that are each numbered with a different positive integer from 1 through 10

Problem Solving — algebra and arithmetic (GMAT Focus Edition)
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by mensanumber » Sun Nov 22, 2020 5:09 pm
The only contents of a container are 10 disks that are each numbered with a different positive integer from 1 through 10, inclusive. If 4 disks are to be selected one after the other, with each disk selected at random and without replacement, what is the probability that the range of the numbers on the disks selected is 7 ?

A) 1/7

B) 3/14

C) 2/7

D) 1/2

E) 15/28
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Re: The only contents of a container are 10 disks that are each numbered with a different positive integer from 1 throug

by mapadvantageprep » Sat Mar 02, 2024 11:59 pm
The pairs that can give the range of 7: (8, 1), (9, 2), and (10, 3).

For each of these pairs, the number of ways to choose the remaining two numbers from the six numbers that are between the smallest and largest in each case (e.g., for the pair (8, 1), choosing any two from (2, 3, 4, 5, 6, 7)) is calculated using combinations: C(6, 2).

Thus, the total number of favorable outcomes (where the range is exactly 7) is 3 (for each pair of extremes) multiplied by C(6, 2) (for the middle numbers).

The total number of possible outcomes when selecting 4 disks from 10 without replacement is given by C(10, 4).

Let's calculate the required probability based on this approach.

The probability that the range of the numbers on the disks selected is 7 is:

​ \(\frac{3x6C2}{10C4}=\frac{45}{210}=\frac{3}{14}\)

The correct answer is B, 3/14


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