Question

Simple answers but work shown thank you!

(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)

Number of distinct letters in the word wombat is 6.

Since the ordering of letters are important, number of different ways the letters of the word wombat be arranged = ⁶P₆

= 6! / (6 - 6)!

= 6! / 0!

= 6 * 5 * 4 * 3 * 2 * 1

= 720

(b)

If the letters wo must remain together (in this order), we can treat wo as a single letter. So, the number of distinct letters became 5.

Number of different ways the letters of the word wombat be arranged if the letters wo must remain together (in this order) = ⁵P₅

= 5! / (5 - 5)!

= 5! / 0!

= 5 * 4 * 3 * 2 * 1

= 120

(c)

Since the ordering of letters are important, number of different ways 3-letter words can be formed from the letters of the word wombat = ⁶P₃ = 6! / (6 - 3)!

= 6! / 3!

= 6 * 5 * 4

= 120

If w must be the first letter of any such 3-letter word, we need to choose 2 letters from the remaining 5 letters.

Number of different ways = ⁵P₂ = 5! / (5 - 2)!

= 5! / 3!

= 5 * 4

= 20