Question

A word is to be formed using some or all of the 8 letters.
S U N S H I N E
Find the total number of ways of forming the word if
a) The word is 8-letter long. The letters can be in any combinations.
b) The word is 8-letter long. The word must begin with a consonant and end with a vowel.
c) The word is 2-letter long.
d) The word is 6-letter long. All the letters must be different.

Answer 1

a)

$${8!\over 2!\cdot 2!}=\dfrac{1\cdot2\cdot3\cdot4\cdot 5\cdot6\cdot7\cdot8}{1\cdot2\cdot1\cdot2}=10080$$

b) If it must start with S

$$1\cdot{6!\over2!}\cdot3=1080$$

If it must start with N

$$1\cdot{6!\over2!}\cdot3=1080$$

If it must start with H

$$1\cdot{6!\over2!\cdot2!}\cdot3=540$$

$$1\cdot{6!\over2!}\cdot3+1\cdot{6!\over2!}\cdot3+1\cdot{6!\over2!\cdot2!}\cdot3=2700$$

c)

$${8!\over 2!\cdot 2!(8-2)!}=8$$

d) 6 different letters

$$6!=720$$