Trying to Unlock a Cell PhoneDate: 11/16/2003 at 12:45:46 From: Savannah Subject: four number combination to a locked cellular phone I put in the local newspaper that I found a cellular phone on the side of the road. No one answered the ad so now I thought I would use it but it's locked by a four digit number. I've tried a lot of combinations and can't seem to get the right one. Can you help? Savannah Date: 01/05/2005 at 14:52:55 From: Doctor Douglas Subject: Re: four number combination to a locked cellular phone Hi Savannah. Although we can't tell you what the code is to unlock the phone, we can compute the number of possible combinations (thus giving you some information about how long the task might take). The number of possibilities for the initial digit is 10, because the initial digit could be any of {0,1,2,3,4,5,6,7,8,9}. For exactly the same reason, the number of possibilities for the second digit is 10. This means that the number of possible combinations for *just the first two* digits is 10 x 10 = 100. In fact, it's not too hard to list the possible sets of two consecutive digits. Here's the list, where I use dots (...) to save me from writing out every element of this somewhat boring list: {00,01,02,03,...,08,09,10,11,...,18,19,20,21,22,...,97,98,99}. You can see that these are just the numbers from zero to 99, where we include leading zeroes. Now let's add the third digit. There are ten possibilities here, and these combine with each of the hundred possibilities for the first two digits: {000,001,002,...,997,998,999}. There are 10 x 10 x 10 = 1000 combinations in this list. You can probably see the pattern now: for a sequence of N symbols, each of which has K possibilities, the number of possible combinations is K x K x ...x K = K^N \__________/ there are N "K to the power N" factors of K in this product You can apply this formula to calculate the number of four-digit PIN combinations. This formula is a direct result of the "Fundamental Principle of Counting", which can be paraphrased as "If there are P ways of doing one thing and Q ways of doing another, then there are P x Q ways of doing both". I should mention that although I have used the word "combinations" here (in keeping with its usage with respect to locks and locking), the word "combination", as used by most mathematicians, has a different meaning: a way of choosing a subset of some elements from a larger set, without regard to ordering or sequence. For example, choosing a set of three representatives from a group of fifteen. For more information about these sorts of possibilities, you can check out the following web page: Ask Dr. Math FAQ: Permutations and Combinations http://mathforum.org/dr.math/faq/faq.comb.perm.html In your problem above, each of the four elements IS distinct from each other (i.e. knowing the first digit of the code is completely independent from knowing the last digit of the code). Your problem can be thought of in these terms as a "permutation with replacement", as illustrated by the following answer in our archives: Combinations and Permutations with Replacement http://mathforum.org/library/drmath/view/64626.html Our archives also have many problems that illustrate the use of the Fundamental Principle of Counting (counting-by-multiplication): Combinations of Three Words http://mathforum.org/library/drmath/view/56148.html Path Possibilities http://mathforum.org/library/drmath/view/56208.html Maximum Possible Combinations http://mathforum.org/library/drmath/view/56130.html Combinations of Images on a TV Screen http://mathforum.org/library/drmath/view/56108.html Counting Answer Keys http://mathforum.org/library/drmath/view/56112.html Examples of the Fundamental Counting Principle http://mathforum.org/library/drmath/view/54298.html I hope that this helps you with your cell phone. If you decide that there are too many combinations to try each one individually, I suggest that you take the phone to one of the service providers that use that phone. They usually have special equipment that can find out to whom the phone is registered, or if it is your own phone, to "reset" it so that it is usable. - Doctor Douglas, The Math Forum http://mathforum.org/dr.math/ |
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