I'm graduating with a Bachelor of Science in Mathematics this year, so hopefully I can help you with your questions.
1) How many unique 10-character password exist, if valid characters consist of only lowercase letters and numbers, and at least one character in the password must be a number?
1. It's a combinatorics (counting) problem. You basically calculate all the possible passwords that can be generated with 36 (26 letters + 10 digits) characters and then subtract the total number of passwords that consist of only letters. So the answer is 36^10-26^10. It's a lot easier than using summations and combinations to add up the sum of all possible cases like Edison did. Edison, you make something easy.... so complicated
I have my friends that are like that.
2. Going to disagree with the answer. It should be about 7.959%, not almost 0%. Each round, you have a 50-50 chance, so if you play a total of 100 rounds, then there are 2^100 unique possible outcomes, each equally likely with a probability of 1/(2^100). Since the person wins exactly 50 times, you have the multiply 1/(2^100) by the number of ways you can get exactly 50 wins out of 100 rounds, which is isomorphic to a combination problem. So the answer is 1/(2^100) * 100C50
Answer in LaTeX generated by mathurl.com:
3. Done by Edison
4. Done by Edison.
5. Going to disagree with Edison. It should be higher than 1/6. It's a Game Theory question where knowing a piece of information will dynamically change the probability. It also has basis in probability theory. I computed it to be 18.18%.
6. Done by Edison
7. Done by BFM_NavyJHk... backward.
8. @Racerrr.... there's not really a "formula" but it has only one simple constraint so it's easier to just set Fx=Fy=Fz=0 and solve for x,y,z instead of using Lagrange Multipliers. There are other easier methods if you ever take a course in Optimization Theory. If you still need a step-by-step guidance for this problem or any other, feel free to let me know.
9. I computed it to be 49 zeroes.... Number Theory certainly helps.
I'm still a n00b in math though, so if anyone thinks any of my answers are incorrect, then let me know.