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

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

Quote:

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.

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

Quote:

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.

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

Quote:

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