Bags, Balls and colours

Probability Level 2

In a bag, there are some balls of the same size that are colored by $7$ colors, and for each color the number of balls is $77$ . At least how many balls are needed to be picked out to ensure that one can obtain $7$ groups of $7$ balls each such that in each group the balls are monochromatic?

Note: Monochromatic means that all balls in the group are same in color. The balls in different groups can have the same color. For example, if we had 49 balls of the first color, then we are done.

The answer is 85.

4 solutions

Karan Taneja
Sep 22, 2014

Assume for a moment that you are the most unlucky person on this earth. What would happen if 'you' are playing this game?

You'll need to pick out this minimum number of balls to get the 7th set of 7 balls having same colour. And until you pick up this last, you won't get those 7 sets of 7 balls.

What else would happen to you?

You'll get 13 balls of any 6 out of 7 colour and 6 balls of the remaining 1 colour. Did you notice how unlucky you are that you didn't get the 7th ball of the 7th set you wanted to create until all the colours have 6 extra balls that are useless to you?

So now you are having 13 times 6 plus 6 balls out of the bag which is 84. And now you are picking up that 85th ball when you complete this set.

(Your are not that unlucky, you are 'brilliant'.)

where is the fact of 77 been used ?

A Former Brilliant Member - 6 years, 8 months ago

Nowhere. They forgot that you need at least 78 balls to ensure that you have 2 balls of different colors. If you pick 77 they could all be of the same color. So you need 77x6+1, to ensure that you picked at least one ball of all colors. Or 77x6+7 = 469 to have 7 groups of 7 balls each such that in each group the balls are monochromatic.

Мирослав Стјепановић - 5 years, 3 months ago

Monochromatic groups need not be different colored groups you see. So 77 is of no need.

Piyushkumar Palan - 3 years, 9 months ago

David Moore
Jun 3, 2016

In this kind of problem, it's sometimes easiest to figure out the configuration with the maximum number of balls such that, whatever ball is picked next will satisfy the condition. So, in this case, we ask, "What configuration gives us the highest number of balls without having 7 monochromatic groups?" Well, we can have 6 monochromatic groups of 7 (group colors irrelevant), plus 7 monochromatic groups of 6 (one group of each color), for a total of 84 balls, and still not satisfy the condition, but the next ball will definitely put us over the top. Thus the answer is 85.

did exactly this

Ajinkya Shivashankar - 4 years, 7 months ago

Sindhuja Reddy
Jan 4, 2019

Let us start like we can choose the ball and we don't want any colour to be repeated. So we will pick 7 different colured balls consequently 6times like that (7x6=42).Anyhow the 43rd ball will be 7th one of either of the colour. Now putting this group of 7aside we start.Thus the removed group colour is preferred for next 6balls and unavoidably the next one(43+7) makes an another group.Now put this group aside.Similarly we do for all the groups and thus 43+(7x6)=85 balls are required.

Madhav Gupta
Sep 7, 2014

at the first step for getting a group of 7 balls with the same color at least and at least 43 balls must have to e picked up to get 7 of the same color so if we pick out 42 there WILL be 6 color balls of the same color definately assuming maximum deviations so after this withdrawal if we again draw 7 balls for getting 43 balls then again we WILL get the same 7 color balls doing this for 6 times we would get the desired.... 43+6*7=85 so answer is 85

Absolutely right! Have you done this before?

Krishna Ar - 6 years, 9 months ago

I don't understand this. If you have 77 balls of each out of 7 colors, that's 539 balls. To be sure that you have at least 1 of each color you need 77x6+1 balls. Or 77x6+7 = 469 balls to be sure you have 7 balls of each color. If you pick 468 balls, you could get all balls of all six colors, and remaining 6 could be of that 7th color.

Мирослав Стјепановић - 5 years, 3 months ago

Why cant the answer be 49.

Manu Sahni - 6 years, 9 months ago

While you could pick 49 balls and have 7 monochromatic groups, you aren't guaranteed it (you can't "ensure" it).

Anonymous Anonymous - 6 years, 9 months ago

nope but i do and study higher math and physics concepts and studied something like pigeon theory i dont remember exactly what it was but i applied it somewhat here

by the way from where do you get such questions???

madhav gupta - 6 years, 9 months ago

Bags, Balls and colours

The answer is 85.

4 solutions

0 pending reports