After 1 lesson on permutation and combination for J2 students taking H2 math, all the students are able to figure out how many ways can Raffles form a 4-letter password for his iPhone using all the letters in his name. We all understand the CMI (Chinese, Malay, Indian) question, we all know how many ways there are to divide 10 people into 5 indistinguishable pairs, we even know many ways to seat 5 people around a round table with 8 chairs around the table at all times, we know how to handle numbered seats, etc, but...
There are some who cannot understand (see) why we need to divide our answer by 2 when we are threading colored beads in a ring!
Typically, the question goes like this: How many ways can 5 different colored beads be threaded around a ring?
The solution is simple, if we were to arrange the 5 colored beads in a circle on a table, the answer would have been (5-1)!=4!=24. Everyone understands and agrees with this.
However, because we're threading them in a ring, which we can hold up and "flip", it is reversible, we need to divide 24 by 2 otherwise, we will be double counting! And therefore, the correct answer should be 24/2=12.
Here comes the trouble! Why must we divide it by 2??
Here comes the enlightenment!
Never thought one of my daughter's favorite toys can become such a useful teaching tool! Of course, everyone understands how the "flipping" of the ring necessitate the dividing by 2!