[Math] How many possible combinations/permutations for 5 letters and 5 numbers

combinationscombinatoricspermutations

I have been given a scenario that includes a generated code that contains 5 alphabetic characters (A-Z) and 5 numbers (0-9). When concatenated, the code looks something like:

ABCDE12345

I have tried splitting them in two and calculating the possible combinations for the letters and numbers separately but I am unsure what to use for n and r.

Best Answer

This problem depends on whether or not you can repeat letters. If you can repeat letters there are $26$ choices for each letter and $10$ choices for each digit. In such a case there are $ 26^5\times 10 ^5 $ ways.

If you can't repeat letters then we have to arrange $5$ letters from $26$ possible letters and arrange $5$ numbers from $10$ numbers (the two choices are independent). In this case there are $ P(26,5)\times P(10, 5) $ ways.

Related Solutions

[Math] How many possible combinations of a HEX string

I think the sanest way to split this is by how many different letters and numbers is contained in the string. Call the total number of different symbols $k$. The possibilities are then:

k=5    1+4  2+3  3+2  4+1
k=6    1+5  2+4  3+3  4+2  5+1
k=7    1+6  2+5  3+4  4+3  5+2  6+1
k=8    1+7  2+6  3+5  4+4  5+3  6+2
k=9    1+8  2+7  3+6  4+5  5+4  6+3
k=10   1+9  2+8  3+7  4+6  5+5  6+4

For each of the 33 entries in the table, the number of ways to select which hex digit appear at all is a simple product of binomial coefficients.

From there each row of the table can be summed and treated uniformly. First select which $10-k$ of the $k$ symbols are going to appear twice, for a factor of $\binom{k}{10-k}$. Then multiply by $10!$ for the ways to arrange the symbols you have selected, and divide by $2^{10-k}$ to account for the fact that each pair of two equal symbols cannot be distinguished.

Combinatorics – Possible Combinations in 8 Character Password

Start with all $8$-character strings: $95^8$

Then remove all passwords with no lowercase ($69^8$), all passwords with no uppercase ($69^8$), all passwords with no digit ($85^8$) and all passwords with no special character ($62^8$).

But then you removed some passwords twice. You must add back all passwords with:

no lowercase AND no uppercase: $43^8$
no lowercase AND no digit: $59^8$
no lowercase AND no special: $36^8$
no uppercase AND no digit: $59^8$
no uppercase AND no special: $36^8$
no digit AND no special: $52^8$

But then you added back a few passwords too many times. For instance, an all-digit password was remove three times in the first step, then put back three times in the second step, so it must be removed again:

only lowercase: $26^8$
only uppercase: $26^8$
only digits: $10^8$
only special: $33^8$

Grand total: $95^8 - 69^8 - 69^8 - 85^8 - 62^8 + 43^8 + 59^8 + 36^8 + 59^8 + 36^8 + 52^8 - 26^8 - 26^8 - 10^8 - 33^8 = 3025989069143040 \approx 3.026\times10^{15}$

Best Answer

Related Solutions

[Math] How many possible combinations of a HEX string

Combinatorics – Possible Combinations in 8 Character Password

Related Question