B3. We must choose a seven-character password that contains at least one letter and at least one digit. We can choose from the characters $A, B, C, G, J, M, R, Z$ and 3.

a) How many possible choices are there for the password if the characters cannot be repeated?

b) How many possible choices are there for the password if the characters can be repeated?

## 16th Mathematics Knowledge Competition for Students of Secondary Technical and Vocational Schools  National Competition, April 16, 2016

## Solutions to Problems and Scoring Guide

(April 3, 2016, $20:53$ )

A competitor who arrives at the solution by any correct method (even if the scoring guide does not provide for it) receives all possible points.

A correct method is considered any procedure that:

- sensibly takes into account the wording of the problem,
- leads to the solution of the problem,
- is mathematically correct and complete.

If an intermediate or final result can be recognized, guessed, read from a diagram, or calculated mentally, the competitor generally receives all the points provided. If, however, the solution is guessed (it is not possible to arrive at it without calculation), it is scored with 0 points.

A competitor who has only partially solved the problem, from otherwise correct solving procedures but without a visible path to the final solution of the problem, cannot receive more than half of the possible points.

The asterisk '*' next to the points means that the point or points can be awarded for a correct procedure, even if the calculation is incorrect.

## First Year

| $\mathrm{A} 1$ | $\mathrm{~A} 2$ | $\mathrm{~A} 3$ |
| :---: | :---: | :---: |
| $\mathrm{D}$ | $\mathrm{E}$ | $\mathrm{E}$ |
B3.
a) Noting that $v$ is a password consisting of 6 letters and one digit ..... 1 point
Determining that there are 7 possible positions for the digit ..... 1 point
Calculating the number of possible arrangements of letters in the password $8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3$ ..... 1 point
Calculating the number of possible choices for the password $7 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3=141120$ ..... 1 point
b) Determining that the total number of possible choices for the password, if there were no restrictions that it must contain at least one letter and one number, is $9^{7}$ or 4782969 ..... 1 point
Determining that the total number of possible choices for the password, if it contained only letters, is $8^{7}$ or 2097152 ..... 1 point
Determining that there is only one password composed entirely of the digit 3 ..... 1 point
Calculating the number of possible choices for the password $9^{7}-8^{7}-1=2685816$ ..... 1 point