Today I did permutations and I read about this theorem:
"The number of permutations of $n$ objects, where $p$ objects are of the same kind and rest are all different $= n!/p!$"
Now this formula is certainly for using up the maximum spaces possible to arrange things.
Like, the word $SCHOOL$ when solved for, by this formula, would give me $6!/2! = 360$. This would give me number of all 5 letter words possible without repetition, from the letters of $SCHOOL$ if I'm not wrong. I was wondering if there is a formula using which, I can find the number of words without repetition, with as many letters as I want.
To put it another way, how can I find out the number of 3-letter words that can be formed from $SCHOOL$ without any repetition? A general formula maybe?
If you have an alphabet $\Sigma = \{a_1,\dots,a_n\}$ of $n$ different symbols and you want to form words of length $k$ with each symbol appearing at most once, the number of these is $$ n\cdot (n-1) \cdot \ldots \cdot (n-k+1) = \frac{n!}{(n-k)!} = \binom{n}{k} k!. $$ In your example the alphabet is $\Sigma = \{\mathrm S,\mathrm C,\mathrm H,\mathrm O,\mathrm L\}$ so $n=5$ and $k=3$.