Answer to Question #98474 in Discrete Mathematics 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). The number of Distinct letters in the word "wombat " = 6

The number of different ways can the letters of the word "wombat" be arranged

$$=P(6, 6)$$

$$=\frac {6!}{(6-6)!} = \frac {6!}{0!} =\frac {6 \times5 \times 4\times 3 \times 2 \times 1} {1} =720$$

(b) The letter "w and o" can treat as one Unit

So, the number letters = 4 letters + one unit of w and 0= 5 things

The number of ways 5 things can be arranged is

$$= P(5, 5) = \frac {5!} {(5-5)!} = \frac {120} {1} = 120$$

(c) The number of different 3-letter words can be formed is

$$P(6,3) = \frac {6!} {(6 -3)!} = \frac {6 \times 5 \times 4 \times 3 \times 2 \times 1} {3!}$$

$$= \frac {6 \times 5 \times 4 \times 3 \times 2 \times 1} {3 \times 2 \times 1} = 6 \times 5 \times 4 = 120$$

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

$$= P(5, 2) = \frac {5!} {(5-2)!} = \frac { 5\times 4 \times 3 \times 2 \times 1} {3!}$$

$$= \frac { 5\times 4 \times 3 \times 2 \times 1} {3 \times 2 \times 1} = 5 \times 4 = 20$$