You’re not taking into account that the digit can occur in any of the six positions. There are $10\cdot36^5$ passwords that begin with a digit. There are also $10\cdot36^5$ passwords that end with a digit; some of these also begin with a digit and have already been counted, but some do not, so your figure of $10\cdot36^5$ is necessarily too small.
The easiest way to count the acceptable passwords is to note that there are $36^6$ six-character strings made up of upper-case letters and digits, and $26^6$ of them are made up entirely of letters, so there are $36^6-26^6$ that include at least one digit.
It is possible to count them directly, but the counting is more complicated. For each of the $6$ positions in the password there are $10\cdot36^5$ passwords having a digit in that position, so to a first approximation there are $6\cdot10\cdot36^5$ acceptable passwords. However, as noted in the first paragraph, this counts some passwords more than once. For each pair of positions in the password there are $10^2\cdot36^4$ passwords having digits in both of those positions, and all of these passwords have been counted twice. Since there are $\binom62$ pairs of positions, we must subtract $\binom62\cdot10^2\cdot36^4$ to get rid of the double-counting. Unfortunately, this overcompensates, and there are further corrections to be made. The net result, given by the inclusion-exclusion principle, is
$$\sum_{k=1}^6(-1)^{k+1}\binom6k10^k36^{6-k}\;.$$
Either way, the result is $1,867,866,560$.
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Try Now Brittany R. I need to know how many passwords can be formed
4 Answers By Expert Tutors
this is a permutation problem there are 26 letters of the alphabet
you want 6-character passwords without repetition
the first time you pick a letter, you pick from 26 letters
the second time you pick a letter, you pick from 25 letters (no repetitions !)
the third time you pick a letter, you pick from 24 letters, and so on
the answer is 26*25*24*23*22*21=165,765,600 passwords
This problem is VERY SIMILAR to the CD problem that I wrote an answer for. Order matters because if I type "faster" as my password, I can't type "astfre" which is all the same letters, but the order isn't correct, so it wont register as correct. How many options of letters do you have for the first letter->26, what about the 2nd-Well we used one already so there are 25 left to choose form. Third letter, 24 choices, 4th letter 23 choices, 5th->22 and 6th 21.
Think of it like that. Those dashes are the places for your letters.
26 * 25 * 24 * 23* 22 * 21
This problem comes from a fun little branch of mathematics called "combinatorics." Basically, they want to know how many different combinations you could make out of the letters of the alphabet, given that they're all lowercase and no repeats are allowed. So let's put six boxes on our piece of paper, one for each character in the password. Got that?
Now, how many possible choices are there for the first character? We know the letters are all lowercase, so it would just be 26, since that's how many letters are in the alphabet. So the first character has 26 choices. Now, move on to the second character. Still 26 letters in the alphabet, but WAIT! We've already used one in the first position, and we know we can't repeat it. So now there are only 25 letters to choose from. Continue in this way to find the number of possibilities for each position in the password. You should end up with:
So now comes the fun part. If we know how many possibilities there are for each position in the password, we simply multiply them all together to get the total number of possible combinations. (There is a proof of why this works, but it's long and complicated so I won't get into it here.) Do that, and you'll get:
26 x 25 x 24 x 23 x 22 x 21 = 165,765,600
So there are 165,765,600 different password combinations using the parameters specified. Hope that helps!
Vivian L. answered • 10/22/13
Microsoft Word/Excel/Outlook, essay composition, math; I LOVE TO TEACH
Thank you for another great question!
26 x 25 x 24 x 23 x 22 x 21
You initially have 26 characters to select from.
Then you have 25 characters to select from.
Then you have 24 characters to select from.
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