# Thread: stuck at this problem for more than two hours...

1. ## stuck at this problem for more than two hours...

It is given that there are at least 1680 ways of painting m different colours into 4 different regions.

Show that m^2 - 3m - 40 >= 0 (Hint: x^2 - 3x + 42 > 0 for all real values of x)

How to show....

have been stuck by this problem for two hours... big headache

2. ## Re: stuck at this problem for more than two hours...

Originally Posted by kennysiu
It is given that there are at least 1680 ways of painting m different colours into 4 different regions.

Show that m^2 - 3m - 40 >= 0 (Hint: x^2 - 3x + 42 > 0 for all real values of x)
What does "at least 1680 ways of painting m different colours into 4 different regions" mean?
Does order matter? Can a colour be repeated? What about rotations.
You have not fully described the question.

3. ## Re: stuck at this problem for more than two hours...

Originally Posted by Plato
What does "at least 1680 ways of painting m different colours into 4 different regions" mean?
Does order matter? Can a colour be repeated? What about rotations.
You have not fully described the question.
order does matter so it is permutation. Colour cannot be repeated. I don't understand what you mean by rotation. Never encounter such a term.

4. ## Re: stuck at this problem for more than two hours...

Originally Posted by kennysiu
order does matter so it is permutation. Colour cannot be repeated. I don't understand what you mean by rotation. Never encounter such a term.
$\displaystyle m(m-1)(m-2)(m-3)\ge 1680$ has $\displaystyle m\ge 8$ as a solution.

Note $\displaystyle m^2-3m-40=(m-8)(m+5).$