Expert: Sherry Wallin - 5/22/2010

Question
question-

a set of three symbal  {a,a,b}.find the the number of way these can be arranged.

aneswar-  c(3,2)=3 ways

Answer
Hi Pratap~
Ask yourself how many ways the first position can be chosen and you would say there are 3 things to choose from so there are 3 ways. Now you have two positions left so you want to know how many ways you can fill the 2nd position and you see that that is 2 because there are only 2 things left to choose from, and finally the third position has just one way to be chosen because there is only one thing left. So there are 6 ways to arrange the 3 things. These are called permutations because you are counting all the ways the 3 things can be arranged, as if you could tell the difference between the two a's. One way to see this is to make one of the a's an * and write all the possible configurations: {*,a,b}, {*,b,a}, {a,*,b}, {a,b,*}, {b,*,a}, {b,a,*}. Now if you couldn't dfferentiate between the a's then {*,b,a} would be the same as {a,b,*} as well as {*,a,b} would be the same as {a,*,b}, and finally {b,*,a} would be the same as {b,a,*} leaving you with 3 combinations. What you have in your answer above is the number of ways you can arrange two of the three things and they are:
{a,a},{a,b}, and {b,a}. There happens to be 3 ways to arrange 2 of the 3 things when you aren't concerned with differentiating between the two a's, because again if you were differentiating between the a's you would have {*,a}, {a,*}, (a,b#, {*,b), #b,*}, {b,a} but notice if you have {*,a} and {a,*} these are just two of the same things just given in a different order.

Summing up what I am telling you is that the number of ways three symbols can be arranged means how many sets of three can you make with the 3 symbols and in your answer you are looking at the number of sets of 2 out of the 3 symbols you can make.

Math Prof