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A password shall begin with 3 different capital letters, followed by 3 to 5 digits between 0 to 9.

a)How many possibilities of passwords are there?

b)How many passwords with the character string "B4" are there?

(with binomial Coefficient = $n \choose k$)

a)For the 3 first digits I have $26\choose 3$ possibilities. Then they are followed by 3 to 5 digits, and I don't know how to approach...

3 Answers 3

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a) There are 3 different cases for this. For when there are 3 numbers, 4 numbers, 5 numbers

There are 3 different letters with 26 to choose from. Hence $\displaystyle\binom{26}{3}$ but this doesn't take into account order so we multiply by $3!$. Then for case 3, 4, 5. Since there are 10 numbers to choose from. We multiply $\displaystyle\binom{26}{3}\times 3!$ by $10^3$, $10^4$, $10^5$ respectively for cases when we use 3, 4, 5 digits. I assumed here that the numbers can be repeated.

b) Since the character string is "B4" (a letter then a number) then B is in the 3rd position, and 4 is in the 4th position.

$_,_,B,4,_,_$ is our format. With permutations $25\times24\times1\times1\times10\times10$

Since B and 4 are fixed we have 1 for the 3rd and 5th position. And $25, 24$ since we alread chose $B$ for the 3rd character.

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    Ok so the solution is 25*24*(1*1)*(10^2+10^3+10^4).I believe after all these helpful comments, I have a much better understanding of the topic.Thank you :)2012-11-22
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a) You can pick the first letter in 26 ways (1 out of 26 letters). The next one will be from the remaining 25 letters, so ways of picking it would be 25. Similarly, the 3rd letter can be picked in 24 ways. So picking the first 3 letters can be done in 26x25x24 ways. For the numbers, ways of picking each digit can be done in 10 ways. As having 3 digits, 4 digits and 5 digits are mutually exclusive events, we can add the ways of generating each. Thus, for 3 digits, we have 26x25x24x$10^3$ ways. Similarly for 4 and 5 digits, giving the total number of ways to be: 26x25x24x($10^3 + 10^4 + 10^5$).

b) You need to fix the 3rd char to B and the first digit to 4 (that is the only place where a number follows a letter). Select the rest from the remaining 25 letters like above, to give the number of ways as: 25x24x($10^2 + 10^3 + 10^4$).

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    @KeithReyes Yep, you are right. Thanks! Fixed.2012-11-22
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a) How about ${10}^3+{10}^4+{10}^5$? (EDIT: Sorry, I first had assumed the digits shold be different as well)

b) Hint: Where can "B4" occur at all?

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    But in a) didn't take into account the permutations for the letters. The sum should be multiplied by $26\times25\times24$2012-11-22