Problem Solving

A buyer buys 3 different items out of the newly introduced 10 different items. If two items were to be selected at random, what is the probability that the buyer does not have both the chosen items?

What does this mean? What do I need to find?

I am not sure, but I will try to answer it.


I think it's asking what's probability that the 3 item he picked, contains the two random chose item.

So it's (number of ways to pick 3 item that contain certain 2 items) /(10c3)

= 8 / (10!/(3!7!))

=8/120

=1/15


OA?

Total number of items = 10
Let S be the sample space, then

n(S) = Numbers of drawing 2 items at random out of 10 = 10C2 = 45

Let E = Event of drawing 2 items at random none of which is the item chosen by that person.

So, n(E) = 7C2 = 21

So P(E) = n(E)/n(S) = 21/45 = 7/15

So answer should be 7/15

@ naylia pls give the answer choices.... or post the original answer.

I also think it's 7/15

Number of ways to choose 2 items out of 10: 10!/2!8!=45
Number of ways to choose 2 items when the 3 items that were bought are out of question: 7!/2!5!=21

So, the P is P of the 2 items that would be chosen out of 7 (when 3 other items are not an option) / P of the 2 items that would be chosen out of all 10 items ---> 21/45=7/15

Here is my take:

the question is asking for the probability that the buyer does not have both the chosen items

The solution posted by codesnooker and Reader is the probability that the buyer does not have any of the chosen items

Not both includes none of the objects but also only one object:

7C2+3*7C1 = 42

So 42/45 = 14/15

It might be easier to obtain the probability that both items have been selected by the buyer p(both) and then take 1 - p(both)

1 - (3C2/10C3) = 1 - (3/45) = 14/15


Or even easier, p(both) without using combinatorics:

Chosen by the buyer: 3 items
Not chosen by the buyer: 7 items

p(both) = p(first object drawn's been chosen)*p(second object drawn's been chosen) = (3/10)*(2/9) = 1/15

Guys, thank you all!!!
This is the OA and explanation:

Number of ways 2 can be selected without taking the three in question. : 7C2
Number of ways 2 can be selected out of 10 = 10C2
Prob. = 7C2/10C2 = 21/45 = 7/15

Hi Nailya,

what's the source? As I said before, I think the cases in which only one of the two selected objects has been chosen by the buyer should also be considered. If the buyer has only one of the objects, it does not have both...

anyone any thoughts?

These is the probability and combinatorics questions set posted by Eric in the Resourses section.

https://www.beatthegmat.com/probability- ... t6206.html