0

How many different words can be formed with word TOMORROW taking all at a time such that all vowels don't occur together??

chahat sharma
  • 11
  • Calculate all coombinations of indexes where $ox$ $ox$ $o$ are put in the phrase and $x$ is whataver comes after $o$, then calulate all permutations of other characters.Multiply both. – Abr001am Feb 12 '18 at 15:46
  • There are $5$ non-vowels. Permuting them gives $5!$ possibilities. There are $2$ $R$'s. So, $5!/2!$ possibilities not counting permuting the $R$'s. For each of them there are $4$ spaces between them to choose $3$ to be the location of the $O$'s. That is $\binom{4}{3}=4$ possibilities. Multiplying you get $4\frac{5!}{2!}$. –  Feb 12 '18 at 15:52
  • @Abra Does that account for words with $oox$ or $xoo$? Looks like the restriction is that you don't have all three together. – John Feb 12 '18 at 16:28
  • @John ouch, that would be even easier to solve! – Abr001am Feb 12 '18 at 16:32

1 Answers1

1

Some hints.

  1. Calculate the number of all possible words you can make with the letters. (Multinomials will help.)
  2. Subtract out the words that have $OOO$ in them. (How many places can $OOO$ occur? Then, how many ways can the rest of the letters fill in the remaining $5$ slots?)

Spolier

From multinomials, the number of words is $8!/(3!2!)$. There are $6$ places to put $OOO$, and $5!/2!$ ways to fill in $TMRRW$. So, the answer is $\frac{8!}{3!2!} - 6 \cdot \frac{5!}{2!}$.

John
  • 26,319