Problem Solutions: Good and Bad

--- tags: mth225 --- # Problem Solutions: Good and Bad ## The problem: This is a subset of exercise 12 from [Section 1.1 on basic counting techniques](http://discrete.openmathbooks.org/dmoi3/sec_counting-addmult.html). >Consider all 5 letter “words” made from the letters $a$ through $h$. (Recall, words are just strings of letters, not necessarily actual English words.) How many of these words are there total? How many contain no repeated letters? And, how many of the words that have no repeated letters also do not contain the word "bad"? ## Some bad solutions All of the bad solutions below actually have right answers, but something is wrong with the solution. ### Bad solution #1 32768, 6720, 6660. :::info **Why it's bad**: The answers happen to be correct but there is no work or narrative to explain why. This isn't a solution at all! It's just answers. ::: ### Bad solution #2 There are eight letters from $a$ through $h$. $8^5 = 32768$ $8 * 7 * 6 * 5 * 4 = 6720$ $6720 = 6720 - 60 = 6660$ :::info **Why it's bad**: This is better because at least the answers have some supporting work. But... + There's nothing explaining why $8^5$ is correct for the first part. Why $8^5$? Why not $8!$ or $\binom{8}{5}$ or something else? The writer should put some English here to explain the choice. + Similarly there's nothing explaining why $8 * 7 * 6 * 5$ is correct for the second part. + In the third part, then number $60$ appears but we have no idea what it is or why it's there or why it's being subtracted. + Also there is a math notation issue here. It says "$6720 = 6720 - 660$" which is not mathematically correct --- those two things are not equal! ::: ### Bad solution #3 There are eight letters from $a$ through $h$. $8^5 = 32768$ $8 * 7 * 6 * 5 * 4 = 6720$ I got those using the Multiplicative Principle. And $6720 = 6720 - 60 = 6660$ because if you subtract out 60 that's what the answer is. :::info **Why it's bad**: Again, better. At least there's an attempt to "explain why". but... + Just saying "I used the Multiplicative Principle" doesn't really help the reader as much as it could. More explanation should be given as to *why* the Multiplicative Principle applies here and *why* the setup for the calcuations are what they are. + We still don't know where that $60$ came from in the third part. + And the sentence at the end does not explain anything --- it merely restates the previous line in English. That basically says "This is the answer because this is the answer." + And there's still the use of the equals sign between two things that are not equal. ::: ## A Good Solution There are eight letters from $a$ through $h$. To form a five-letter word from these, we make a sequence of five choices, each having 8 options. As a diagram it looks like this: | Letter 1 | Letter 2 | Letter 3 | Letter 4 | Letter 5 | | -------- | -------- | -------- | -------- | -------- | | 8 choices | 8 choices | 8 choices | 8 choices | 8 choices | Therefore the Multiplicative Principle applies, and the number of those words is $8 \times 8 \times 8 \times 8 \times 8 = 32768$. If there are no repeats allowed, then we are still making a sequence of 5 choices, but there are 8 choices for the first letter, then 7 for the second because of no repeats, then 6, then 5, then 4. So the number of those words is $8 \times 7 \times 6 \times 5 \times 4 = 6720$. Finally, to count the number of words that have no repeats and also don't contain the word "bad", we'll first count the number of words with no repeats that *do* contain the word "bad". Then we'll take that number and subtract from $6720$, the total number of words with no repeats. This will tell us how many words without repeats *don't* contain "bad". In any word that contains "bad", one of three things must happen: The first three letters are "bad", the second three are "bad", or the last three are "bad". The remaining letters are a free choice, but since there are no repetitions we can't select "b", "a", or "d" and we can't pick the same letter twice. So there are 5 choices for one remaining letter and 4 for the other. The lower three rows in this diagram show what this looks like: | Letter 1 | Letter 2 | Letter 3 | Letter 4 | Letter 5 | | -------- | -------- | -------- | -------- | -------- | | b | a | d | 5 choices | 4 choices | | 5 choices | b | a | d | 4 choices | | 5 choices | 4 choices | b | a | d | In the first row, there are $5 \times 4 = 20$ possible words as a result of those choices. The same is true for the second and third rows. Since these three possibilities have no words in common, this means there are $20 + 20 + 20 = 60$ words that contain "bad". Therefore the number of words without repeats that *don't* contain "bad", is $6720 - 60 = 6660$. :::info **Why this is good:** - First of all, all the answers are correct. - But not only that -- every step leading up to the answers is laid out, in simple language, explaining every step so that the reader doesn't have to do any extra work to fill in missing information. - In other words, the writer is **helping the reader understand** rather than just blasting stuff into a writeup without respecting the reader's need to understand. - The reader might have to stop and think about a step in the explanation. But all the raw material for understanding is here, included in the writeup. - Notice the *strategy* for the third part is stated *before* the solution is begun. The writer here is telling the reader what is about to happen, then explains every step along the way as they make it happen. - Also notice that **the majority of this solution is *not* mathematical computations.** Most of it is words. Generally speaking, **a good solution to a mathematical problem is a combination of words and math**, with math done only when necessary to explain or streamline a thought process. - And notice that the writer here has also given a diagram to help the reader understand the first and third part. If including such a diagram is helpful, do it. (But don't overdo it.) - Finally, notice that the solution is longer than just putting down answers with minimal work; but it's also not *extremely* long. It's pretty short, actually. **A good solution does not have to be lengthy.** In fact you should strive to write in a way that is simple and concise without including any unnecessary material that does not add to the reader's understanding. :::