Math Is Fun Forum

Our local greengrocer had two bags with 17 apples in each. In the first bag, all apples are good, while in the other one, 5 of them are rotten to the core, but seem good to the outside. All the good apples weight the same and all the rotten also weight the same (but have different weight from the good ones). The greengrocer has sold one of the two bags, but without knowing which one! Now he is left with the other bag and wants to find out whether it contains all good apples, or it is the one with the 5 rotten. Using only a balance scale, what is the minimum number of weighings required to determine which bag he has?

During a game show, the host has placed three contestants (A, B, C) in a closed room.  The host places one stamp on each of the contestant’s forehead in such a way that each contestant can only see the other stamps, but not his own.  He places a white or black stamp to the forehead of A, a red or green stamp to B and a yellow or blue stamp to C. He then asks each of them to simultaneously write on a piece of paper the color of the stamp they think that they have on their own forehead. The only choices that are permitted are, either a color, or "I don't know". The contestants will win if at least one of them correctly guesses his own color, but with the restriction that no one must not give a wrong answer (for example, if 2 of the 3 contestants reply "I don't know" and the 3rd one guesses correctly, they win the contest).
The host allows the contestants to discuss and coordinate for a few minutes, before the placement of the stamps, in order to decide on a strategy. What is the optimum strategy that will give them the biggest chance to win?

The sponsor of a basketball team has offered 209 branded t-shirts to the regional high school players. Each of the t-shirts had a unique number from 2 to 210 printed on its back.
In order to randomly assign the t-shirts to the students, the kids were all given a unique number from 1 to 209 and the shirts would be randomly assigned by carrying out some kind of draw. The maths teacher, however, decided to make it a bit more tricky, so he asked the students to find a way to assign the shirts in such a way, that each student's number is a factor of the t-shirt's number. He then asked them to find how many such ways there are and also to give an example of such an arrangement (we remind that each number has at least two factors, 1 and the number itself).

Hi guys,
Anyone knows any site to generate permutations / combinations over the 10000000 limit?
The Combinations and Permutations Calculator under this site is a GREAT tool but unfortunately has this limitation (
I don't want just to calculate them (this is easy) but also to GENERATE them.
Many thanks!!

A young boy is going to school from his house in the countryside and has to pass through a small forest. He has a snack with him, which he must eat at the noon break, but he is hungry so he eats it while walking to school.
He realizes, of course, that there will be nothing left to eat at the break!
Fortunately, the forest is full of pears that have fallen off the trees, one every few steps, on his way to school. Some of them are rotten, though, which cannot be discerned from the good ones. Only one pear can fit in his lunch box, so he must take with him exactly one pear. Furthermore, he has with him a small balance scale which he needs for the class of physics. He knows that all good pears have exactly the same weight and also that all rotten pears have different weights from the good ones (can also be different from each other but not necessarily). Knowing also that there are more good pears than rotten, how can he select exactly one good pear without walking back and forth?