Answer to Question #98517 in Combinatorics | Number Theory for Ahmed

Question

(a) In how many different ways can the letters of the word wombat be arranged?
(b) In how many different ways can the letters of the word wombat be arranged if the letters wo
must remain together (in this order)?
(c) How many different 3-letter words can be formed from the letters of the word wombat? And
what if w must be the first letter of any such 3-letter word?

Answer 1

(a) Order matters in arranging letters of the word WOMBAT, thus permutation is used. n=6, total number of letters, and r=6, total number of letters picked for arrangement. $nPr=\frac{n!}{(n-r)!}$ Thus, $\operatorname{nPr}\left(6,6\right)=\frac{6!}{0!}=720$

(b) For WO to remain together (in that order) during arrangement, the letters are treated as one letter so that WOMBAT is assumed to have 5 letters. Permutation is applied. n=5, r=5, thus, $\operatorname{nPr}\left(5,5\right)=\frac{5!}{0!}=120$

(c) In forming three-letter words from, MOMBAT, order matters, thus permutation is applied. n=6, and r=3. Hence, $\operatorname{nPr}\left(6,3\right)=\frac{6!}{3!}=120$

If W must be the first letter in the three-letter word, the letter is removed in both parameters, n, and r, and permutation is applied. Thus, n=5, and r=2, hence $\operatorname{nPr}\left(5,2\right)=\frac{5!}{3!}=20$