I've been stuck on these for a while. I don't understand what the term linear ordering is for the first one. Would love it if someone can help with these answers, so I can study using it for my final exam! Thanks.
In how many ways can the six letters in RANDOM be arranged in linear order so
that
(i) the two vowels A and O are consecutive?
(ii) the letter A precedes the letter O?
A 30-member mathematics department must select one person to chair the discrete
mathematics committee and two other people to co-chair the calculus committee.
(i) In how many ways can this be done?
(ii) In how many ways can this be done if Professor Blue refuses to chair the
discrete mathematics committee?
(iii) In how many ways can this be done if Professor Green refuses to co-chair the
calculus committee?
Best Answer
For part (i), group the $AO$ together into a single unit. So how many ways can we permute five letters? There are $5!$ ways, correct. We now consider $OA$, which gives us another $5!$ permutations. So there are $2 * 5!$ ways to get $A$ and $O$ next to each other. Part (ii) should be easy to answer from here.
For part (iii), how many ways can you pick the chair? There are $\binom{30}{1} = 30$ ways to choose the chairperson. You then choose the co-chairs in $\binom{29}{2}$ ways. Multiply these together to get $30 * 29 * 14$ ways to choose a committee.
For part (iv), if one person refuses to chair the Discrete committee, we only have $29$ possible chairpersons from which to choose. I think you should be able to answer part (v) on your own, or at least attempt it now.