Quandaries
and Queries 

Hi I am Kevin Cooke I am a parent of four boys aged 8 to 14 years. My question is as follows. My 12 year old has asked me how to figure out the number of four digit combinations possible using 0 thru 9 without repeating a number? Can you help? Regards Kevin 

Hi Kevin, It depends what you mean by combinations. If he is thinking of combinations as in a combination lock where the order is important (that is 2,4,6,5 and 2,4,5,6 are different) then you can proceed this way. Think of constructing the 4 digit combination, reading from left to right. For the fist digit you have 10 choices. Once you have decided on the first digit you have 9 choices for the second digit since you can't repeat digits. Thus for a two digits arrangements you have
Now that you have used two digits you have 8 choices for the third digit and hence
Finally there are
Mathematicians however use the word combinations for arrangements where order is unimportant. That is (2,4,6,5); (2,4,5,6); (2,5,4,6); (6,4,2,5);... are all the same combination of the four digits 2, 4, 5 and 6. In fact, using the same development as above you can see that there are
different orders of the digits 2, 4, 5 and 6. Hence in the list of 5040 arrangements above every four digit combination is repeated 24 times. With this meaning of combination there are
There is an expression for this using the factorial notation. For a positive integer n, n factorial, written n!, is defined by
So, for example
With this notation the number of combinations of 10 digits taken 4 at a time is
In general the number of combinations of n things taken k at a time is
Penny 

