$$ \binom{12}6 = \frac{12\cdot11\cdot10\cdot9\cdot8\cdot7}{6\cdot5\cdot4\cdot3\cdot2\cdot1} = 924. $$ Sometimes it's hard to talk students out of computing both the numerator and the denominator in this expression and then dividing.
I can think of three reasons for this:
- It takes at least some effort to learn that one can simplify things like this by canceling.
- Elementary school pupils are told $3\times4$ means "multiply $3$ by $4$", i.e. "$\times$" is a verb in the imperative mood. Even in elementary school, I reminded myself every time I heard this that $3\times4$ is a noun, but I've never seen any evidence of anyone else doing that.
- Calculators are anesthetics. Students approach math problems with great anxiety, and, confronted with an expression like the one above, frantically reach for their calculators the way a drowning person reaches for anything that floats. A calculator WILL get the student out of the terrible predicament that is a math problem and calculators are INFALLIBLE. If $8/3=2.666666\ldots$ and a calculator says $2.666\times51=135.966$ then that is EXACT and INFALLIBLE. Whoever says the answer is $136$ is rounding it. If $\pi=3.14$ and you find that $3.14^3\cong30.959$, and you want to be more accurate, just add more of the digits that the calculator shows you when you find $3.14^3=30.959144$, and if the exam question says you are to give an EXACT answer, that means give all the digits that your calculator will show you. Calculators COMPLETELY OBVIATE ALL NEED TO THINK ABOUT ANY QUESTIONS THAT THIS PRESENT PARAGRAPH MIGHT SUGGEST. Anyone who doubts that is a lunatic.
In simplifying rational functions, plainly one should cancel first, but with rational numbers, it's not always clear to student what the advantage is, if any. Are there examples not involving algebra, but only arithmetic, that are as cogent to newbies as are examples of simplifying rational functions?