pigeonhole principle (1 Viewer)

[Masaken]

How would I approach this question? (Not asking for a solution but rather how to start answering this question -- what do I look for? What do I consider? That sort of thing. My teacher explained it but I was so confused the entire time -- she just put up random numbers and circled them.)

A box contains 6 red, 8 green, 10 blue, 12 yellow and 15 white balls. What is the minimum number of balls we have to choose randomly to ensure that we have nine balls of the same colour?

 

[cossine]

How would I approach this question? (Not asking for a solution but rather how to start answering this question -- what do I look for? What do I consider? That sort of thing. My teacher explained it but I was so confused the entire time -- she just put up random numbers and circled them.)

A box contains 6 red, 8 green, 10 blue, 12 yellow and 15 white balls. What is the minimum number of balls we have to choose randomly to ensure that we have nine balls of the same colour?

Just consider the worst case. In this case it would 8 balls of the same color or each color. Of course you will only have 6 balls for red.

 

[Masaken]

Just consider the worst case. In this case it would 8 balls of the same color or each color. Of course you will only have 6 balls for red.

So would it just be 6 + 8 + 8 + 8 + 8 = 38?

 

[cossine]

So would it just be 6 + 8 + 8 + 8 + 8 = 38?

Yes just plus 1

 

[Masaken]

Yes just plus 1

Why would it be plus 1?

 

[yanujw]

Why would it be plus 1?

38 is the maximum amount of ways in which the condition can be avoided. By picking another ball (making a total of 39) you ensure that the condition must be satisfied.

 

[Masaken]

38 is the maximum amount of ways in which the condition can be avoided. By picking another ball (making a total of 39) you ensure that the condition must be satisfied.

Alrighty, thank you, that makes sense!

Is it alright if y'all could help with approaching this question too?

25 students attend a class reunion and shake hands with each other. If no students shake hands with the same person twice, explain why at least two students will have shaken the same number of hands.

 

[yanujw]

Alrighty, thank you, that makes sense!

Is it alright if y'all could help with approaching this question too?

25 students attend a class reunion and shake hands with each other. If no students shake hands with the same person twice, explain why at least two students will have shaken the same number of hands.

Think about the range of values for which someone can shake someone's hand. For any person, they may shake 1,2,3...23 hands. But , if one person shakes 0 hands, there cannot be a person that shakes 24 hands, and vice versa (because there is a contradiction between one person not shaking anyone's hand, and one person shaking everyone's hand). So there are 24 numbers for which a person can perform handshakes.

24 potential numbers of handshakes for a person and 25 people. It's easy enough to do the PP from there.

 

[Masaken]

Think about the range of values for which someone can shake someone's hand. For any person, they may shake 1,2,3...23 hands. But , if one person shakes 0 hands, there cannot be a person that shakes 24 hands, and vice versa (because there is a contradiction between one person not shaking anyone's hand, and one person shaking everyone's hand). So there are 24 numbers for which a person can perform handshakes.

24 potential numbers of handshakes for a person and 25 people. It's easy enough to do the PP from there.

Thanks very much (and to cossine)! Your help is greatly appreciated, I've been struggling with this for a while haha