An urn contains a number of colored balls, with equal numbers of each

Question

An urn contains a number of colored balls, with equal numbers of each color. Adding 20 balls of a new color to the urn would not change the probability of drawing (without replacement) two balls of the same color. How many balls are in the urn? (Before the extra balls are added.)

(A) 210
(B) 200
(C) 190
(D) 170
(E) 95

Are You Up For the Challenge: 700 Level Questions

(A) 210
(B) 200
(C) 190
(D) 170
(E) 95

Are You Up For the Challenge: 700 Level Questions

Kinshook · Answer

Given: An urn contains a number of colored balls, with equal numbers of each color. Adding 20 balls of a new color to the urn would not change the probability of drawing (without replacement) two balls of the same color.

Asked: How many balls are in the urn? (Before the extra balls are added.)

Total balls should be multiple of 20.

IMO B

Asked: How many balls are in the urn? (Before the extra balls are added.)

Total balls should be multiple of 20.

IMO B

abhijith04 · Answer

Bunuel   GMATinsight   KarishmaB   chetan2u  Could someone please help with a detailed solution to get the OA.

Regor60 · Answer

See attachment. Number of balls before should be 190, not 170.

Posted from my mobile device

lg800 · Answer

Could someone please explain how it is 170?

Tanisha2819 · Answer

Kindly provide its solution  Bunuel . This is a very tough question. Couldn't solve it even after spending 10 mins on it

Flower_Child · Answer

Same. Even I got 190.

Bunuel · Answer

The OA is 190. Fixed the OA. Thank you!

Flower_Child · Answer

Consider there are  'n'  different colors.
Consider there are  'y'  balls of each color .

Total balls =  ny

no of ways to draw 2 balls of same color = no of ways to draw 2 balls of first color + no of ways to draw 2 balls of second color + .... n times 
=  y (y-1) + y (y-1) + .... n times  
=  ny(y-1)

no of ways to draw any two balls =  ny ( ny-1)

Probability ( to draw 2 balls of same color) =  \frac{ny(y-1)}{ny ( ny-1) } = \frac{y-1}{ny-1}  --------- (1)

Now , 20 balls are added.

Now , no of ways to draw 2 balls of same color = no of ways to draw 2 balls of first color + no of ways to draw 2 balls of second color + .... n times  +  no of ways to draw 2 balls of the  NEW  color 
= ny(y-1) + 20(19)

no of ways to draw any two balls =   (ny+20) ( ny+20 -1)   =   (ny+20) ( ny+19)

Probability ( to draw 2 balls of same color) =  \frac{ny(y-1) + 20(19)}{ (ny+20) ( ny+19) }   -------- (2)

Now given (1) = (2)

\frac{y-1}{ny-1} = \frac{ny(y-1) + 20(19)}{ (ny+20) ( ny+19) }

\frac{y-1}{ny-1} = \frac{ny^2 - ny +380}{ n^2 y^2 +39ny +380 }

Cross Multiplying

n^2 y^3 + 39 ny^2 + 380y - n^2 y^2 - 39ny -380 = n^2 y^3 - n^2 y^2 +380 ny - ny^2 + ny -380

many terms will cancel out

40 ny^2 - 40 ny = 380 (ny -y) 
 40 ny ( y-1 ) = 380 y(n-1)

2n(y-1) = 19 (n-1)

2ny - 2n = 19n -19

19 = 21n - 2ny

19 = n (21-2y)

We know that n and y has to be greater than 0 and integers since they are number of colors and number of balls respectively.

Since 19 is prime , only two cases exist :

Case 1 :  n=1  
=>  y= 1  
This can't be possible as in this case probability of selecting two balls of same color would be 0 .
Probability before and after adding 20 balls is same . Since probability can't be 0 after adding 20 balls of new colors  this case is not possible  .

Case 2 :  n=19 
 => 21- 2y = 1 
=> y = 10 
This is our answer . Total balls =  ny = 190

bumpbot · Answer

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.

An urn contains a number of colored balls, with equal numbers of each

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

An urn contains a number of colored balls, with equal numbers of each

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards