A cooperative of small farmers has created its own 6-digit numbering system so that consumers can identify the origin of the products:

I) The first 2 digits indicate the sub-region, which are 27, numbered from 01 to 27.

II) The next 2 digits refer to the producer, with each sub-region having 40, numbered from 31 to 70 (note that there is a producer 35, for example, in sub-region 01, 02, 03... etc., i.e., there are 27 producers number 35).

III) The last 2 digits refer to the type of product, which are 28, numbered from 71 to 98.

For example, the number 074197 corresponds to a product from sub-region 07, producer 41 of this sub-region, and type 97.

a) Carlos wrote down the number in the wrong sequence, swapping the 3 parts of the code, writing 900950. Is it still possible to identify the sub-region, producer, and type of the product?

b) In total, how many different codes can be generated, so that they meet the cooperative's conditions?

c) Júnior is a superstitious consumer. He hates the number 15 and does not buy any product that contains it. How many product codes are in this situation?
Solution

a) It is possible, as the numbering of each of the three elements is different. Carlos's product is from sub-region 09, Carlos is producer 50 of this sub-region, and the type of product is 90.

b) There are three elements that make up the code. In the first, there are 27 options, in the second, there are 40, and in the third, there are 28. Thus, the total number of possible codes is $27 \cdot 40 \cdot 28=30,240$.

c) Fixing 15 in the first two digits, there are $40 \cdot 27=1,080$ possibilities. Fixing 15 in the second and third positions of the code, there are 3 possibilities for the $1^{\text{st}}$ digit (0, 1, or 2), 10 for the $4^{\text{th}}$, and 28 for the last two, that is, there are $3 \cdot 10 \cdot 28=840$ possibilities. Since 15 cannot appear in other positions, we have, therefore, $1,080+840=1,920$ product codes in this situation.