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(a) To log on to a certain computer account, the user must type in a 3-letter password. In such a password, no letter may be repeated, and only the lower case of a letter may be used. How many such 3 -letter passwords are possible? (There are 26 letters in the alphabet.)

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(b) From a collection of 52 store customers, 2 are to be chosen to receive a special gift. How many groups of 2 customers are possible?

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(a) To log on to a certain computer account, the user must type in a 33-letter password. In such a password, no letter may be repeated, and only the lower case of a letter may be used. How many such 33 -letter passwords are possible? (There are 2626 letters in the alphabet.)\newline \square \newline(b) From a collection of 5252 store customers, 22 are to be chosen to receive a special gift. How many groups of 22 customers are possible?\newline \square

Full solution

Q. (a) To log on to a certain computer account, the user must type in a 33-letter password. In such a password, no letter may be repeated, and only the lower case of a letter may be used. How many such 33 -letter passwords are possible? (There are 2626 letters in the alphabet.)\newline \square \newline(b) From a collection of 5252 store customers, 22 are to be chosen to receive a special gift. How many groups of 22 customers are possible?\newline \square
  1. Calculate Password Possibilities: Calculate the number of possible 33-letter passwords where no letter repeats and only lowercase letters are used. There are 2626 choices for the first letter, 2525 for the second (since it can't repeat the first), and 2424 for the third (can't repeat the first two). Calculation: 26×25×2426 \times 25 \times 24.
  2. Verify Password Calculation: Check the calculation for any errors. 2626 choices for the first letter, 2525 for the second, and 2424 for the third are correctly considered without repetition.
  3. Calculate Customer Selection: Calculate the number of ways to choose 22 customers from a group of 5252. This is a combination problem, not permutation, since the order of selection doesn't matter. Calculation: 5252 choose 22, which is rac{52!}{2! * (52-2)!}.
  4. Simplify Customer Selection Calculation: Simplify the calculation for choosing 22 customers. 52!2!×50!\frac{52!}{2! \times 50!} simplifies to 52×512×1=1326\frac{52 \times 51}{2 \times 1} = 1326.
  5. Check Customer Selection Calculation: Check the calculation for any errors. The formula for combinations (nk=n!k!(nk)!)n \choose k = \frac{n!}{k! \cdot (n-k)!} is correctly applied, and the arithmetic is correct.

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