Question

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?

1.   469
2.   539
3.   85

Comments

i am not getting  :(

---

Suppose there are 7 colours c1,c2 ....c7 and groups g1,g2,.....g7

Now after 43 draws we will definitely get 1 group with 7 balls having same colour and lets consider this group be g1 and colour c1. So it will be like

7 6 6 6 6 6 6

now if we draw ball of colour other than c1 then we will complete one more group but then it will not be worst case. So we will draw c1 again 7 more time, so that now 2 groups will be completed.

Now again if we draw a ball with colour other than c1 then again a group will be completed becoz we still have 6 balls of each colour c2,c3,....c7. So again we will draw c1 7 times.

This loop will continue till all 7 groups are completed with the colour c1

so number of draws = 43 + 7*6 = 85

---

Actually pigeonhole will not be used here .It takes the worst cse scenario but here we have asked "atleast" so take the best cases.

Answer 1

For, 7 groups of 7 balls we need 49 balls atleast
so we draw randomly 49 balls
assume worst case that minimum number of groups are formed

color	c1	c2	c3	c4	c5	c6	c7
balls	7	6	6	6	6	6	6
balls	6

(1 group and 49 balls)

this is the worst case that we got one group
when we pick next ball it must belong to one of the 7 group(let it be c2)

color	c1	c2	c3	c4	c5	c6	c7
balls	7	7	6	6	6	6	6
balls	6

(2 groups and 50 balls)

when we pick next 6 balls we didnt get any new group

color	c1	c2	c3	c4	c5	c6	c7
balls	7	7	6	6	6	6	6
balls	6	6

(2 group and 56 balls)

when we pick next ball it must belong to one of the 7 group(let it be c3)

color	c1	c2	c3	c4	c5	c6	c7
balls	7	7	7	6	6	6	6
balls	6	6

(3 groups and 57 balls)

when we pick next 6 balls we didnt get any new group

color	c1	c2	c3	c4	c5	c6	c7
balls	7	7	7	6	6	6	6
balls	6	6	6

(3 groups and 63 balls)

when we pick next ball it must belong to one of the 7 group(let it be c4)

color	c1	c2	c3	c4	c5	c6	c7
balls	7	7	7	7	6	6	6
balls	6	6	6

(4 groups and 64 balls)

when we pick next 6 balls we didnt get any new group

color	c1	c2	c3	c4	c5	c6	c7
balls	7	7	7	7	6	6	6
balls	6	6	6	6

(4 groups and 70 balls)

when we pick next ball it must belong to one of the 7 group(let it be c5)

color	c1	c2	c3	c4	c5	c6	c7
balls	7	7	7	7	7	6	6
balls	6	6	6	6

(5 groups and 71 balls)

when we pick next 6 balls we didnt get any group new group

color	c1	c2	c3	c4	c5	c6	c7
balls	7	7	7	7	7	6	6
balls	6	6	6	6	6

(5 groups and 77 balls)

when we pick next ball it must belong to one of the 7 group(let it be c6)

color	c1	c2	c3	c4	c5	c6	c7
balls	7	7	7	7	7	7	6
balls	6	6	6	6	6

(6 groups and 78 balls)

when we pick next 6 balls we didnt get any group new group

color	c1	c2	c3	c4	c5	c6	c7
balls	7	7	7	7	7	7	6
balls	6	6	6	6	6	6

(4 groups and 84 balls)

Then 85th ball and we get 7 monochromatic groups of 7 balls each

Comments

thats what i am asking ,after drwaing 43 balls,
we get first group then you are saying that further we draw 6 balls then it might be the case that they dun belong to any new grp and just to the first group,hence,no new grouo formed.but then you are saying after drawing one more ball,we will definitly get a new group..how??it might be the case that this one draw belongs to first group only.hence still,no new group formed.

---

groups after drawing 6 balls are

(7+6)  6  6  6  6  6  6 ==> 49 ball

now here we have 7 empty groups of 7 different color you are assuming 6 empty groups ...but we have 7 empty group caz monochromatic is given which means each group can have only same color balls ...but it didnt say that only 1 group can be formed from 1 color all 7 groups can be of same color

---

i think you are not getting what i am trying to ask.

i am just asking that how do u know that after drawing 'one' ball(after drawing 49 balls) that the new ball will definitly have a colour other than the first group(which is already formed) and thus it will make up 7 balls for the new group.

Answer 2

Worst case to pick up the balls is like:
(Only if I presume all groups must be of different colors, otherwise 85) 
A1,A2,A3,...,A7,A8,...,A77,

B1,B2,B3,...,B77,

C1,C2...C77,

D1,D2,...D77, ...

...G1,...,G7 .

Which is we encountered balls sequentially.

For first 6 groups, balls picked up = 77*6 =462.

Now for last group we simply pick up 7 more balls.

Therefore total balls removed = 469.

Comments

Yeah I know a group can have seven balls but they are asking about total number of pickings.

---

@Rohit according to question we have to form 7 groups and each group should contain 7 balls of same colour. But in question it is not written that these groups will contain different colour of balls than the other group.

while u are finding the answer considering the colour of balls in a group are different in colour from other group balls.

---

Yes @Digvijay was finding presuming 7 different color groups. Thanks.