How to solve this question

Question

If seven colors are used to paint 50 bicycles then which of the following statements need not be true?

• A. at least eight bicycles are of the same color

• A. at least seven bicycles are of the same color
• B.  at least nine bicycles are of the same color
• C. at most eight bicycles are of the same color

Comments

Please explain your answer

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ceil(50/7)=8  . So option a

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Use pigeon hole

Let there be 7 holes of 7 different colour

The start putting bicycles into these holes one by one

You can fill 49 bicycles with 7 bicycles in each box

But then when you'll put 1 more bicycle one box will have 8 and other boxes will have 7

But none of the boxes have 9

So C is not true always

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consider 7 colours as 7 pigeonholes. now start assigning every bicycle to a pigeon hole serially .After assigning 49 bicycles u will have 7 pigeon holes with each having the height of 7 .so the 50th Bicycle needs to be assigned to one of the pigeon holes so 1 of them would have a height of 8. So here 8 bicyles will have the same color.So in every scenario minimum 8 bicyles will have the same colour

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Use the pigeonhole principle.

==> 50 pigeons and 7 pigeonhole.

For at least =  $\left \lceil \frac{50}{7} \right \rceil = 8$

For atmost  = $. \left \lfloor \frac{50}{7} \right \rfloor = 7$

So, option C and D are wrong.

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@kumar.dilip

You did opposite for at most and at least. Isn't it?

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2. at least seven bicycles are of the same color
3.  at least nine bicycles are of the same color
4. at most eight bicycles are of the same color

Are wrong. A is true. makes B and C wrong. and in D. we can have atmost 50 bicycles of the same color. as there are MAX 50 bicycles. so D wrong. too.