How many strings are there of four decimal digits that:

Question

I don't know which are decade digits (I suppose they are: 10, 20, 30, 40,....etc, but I'm not sure)

a) The same digits can't repeat twice

b) The string contains exactly 2 even digits

Answer 1

These are all straightforward applications of the rule of product which paraphrased says something like this:

To count the number of outcomes or arrangements in a scenario, if you can perfectly describe each such outcome via a sequence of steps such that the number of options at each step does not depend on any previously made selections (though which options are available may change) and every outcome is achieved by exactly one unique sequence of selections, then the total number of outcomes is the product of the number of choices for each step.

I will solve two for you, part (a) and part (b) and leave the rest for you to do.

---

Part (a) asks us to find the number of strings of length $4$ using digits such that no digit is repeated. To do this, we approach via rule of product by describing the sequence of steps and noting the number of options available for each step.

• Choose the first digit (10 options)
• Choose the second digit (9 options)
• Choose the third digit (8 options)
• Choose the fourth digit (7 options)

The total is then the product of these, $10\times 9\times 8\times 7$.

Note, depending on what was chosen for the first digit, the available options for the second digit will change so as to not include what was used for the first. The number of options however will always be nine regardless which exact choice was made for the first digit.

Note also, we could have specified these steps in a different order, for example by choosing the digits in reverse order by starting with the fourth digit, or any number of other ways. There will often be many different equally correct approaches to a problem which may even lead to different expressions for the final result. Assuming everything was done correctly for both approaches, despite the expressions looking different this would imply that the expressions are in fact equal and is a useful tool for proving identities in combinatorics.

---

Part (b) asks us to have the string contain exactly two even digits.

• Pick the locations used by the even digits ($\binom{4}{2}=6$ options)
• Pick which digit is used in the left-most space designated for even digits ($5$ options)
• Pick which digit is used for the remaining even digit ($5$ options)
• Pick which digit is used for the left-most space designated for odd digits ($5$ options)
• Pick which digit is used for the remaining space ($5$ options)

Multiplying these gives us a total of $\binom{4}{2}\times 5\times 5\times 5\times 5$ total outcomes.

Note, $\binom{n}{r}$ is the notation used for the binomial coefficient and is a common counting tool that you should become intimately familiar with.