Answer to Question #262366 in Discrete Mathematics for zid

Question

(a) In how many different ways can the letters of the word donkey be arranged?

(b) In how many different ways can the letters of the word donkey 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 donkey? And what if d must be the first letter of any such 3-letter word?

Answer 1

Let us use the formula for the number of permutation: $P_n=n!.$

(a) It follows that the letters of the word "donkey" can be arranged in $6!=720$ different ways.

(b) Since in this task the string "do" can be thought as a unique construction, it follows that the letters of the word "donkey" can be arranged if the letters "do" must remain together (in this order) in $5!=120$ different ways.

(c) By combinatorial product rule, the number of different 3-letter words that can be formed from the letters of the word "donkey" is equal to $6\cdot 5\cdot 4=120.$

If "d" is the first letter of any such 3-letter word, then by combinatorial product rule, the number of different 3-letter words that can be formed from the letters of the word "donkey" is equal to $5\cdot 4=20.$