math help me plz

Question

Higgins was asked to make a unique three-letter arrangement using only letters from the word home. He can use each letter only once. (His arrangement doesn’t need to be a valid word.)

 

he probability that Higgins forms a three-letter arrangement with vowels as the second and third letters is ?

The probability that Higgins forms a three-letter arrangement with two consecutive consonants is?

Answer 1

The total possible arrangements is given by:

n!                4!                 24

______   = _______  =  ______ =    4

r!(n-r)!       3!(4-3)!         6(1) 

 

The probability of a three letter word with vowels as the second and third letters is the cumulative probability of each letter

for the first letter 2/4 letters are not vowels

for the second letter 2/3 letters are vowels

for the third letter 1/3 letter are vowels

so the total probability is the sum of the probabilities or (2/4) * (2/3) * (1/3) = 4/24

 

The probability of a three letter word with two consecutive consonants the cumulative probability of each letter.

for this problem you have two possible solutions. The first letter can be a vowel and the second and third consonants or

the first and second letters consonants and the third letter a vowel.

 

for the first letter 2/4 letters are consonants
for the second letter 2/3 letters are consonant
for the third letter 2/2 letter are vowels
so the total probability is the sum of the probabilities or (2/4) * (2/3) * (2/2) = 8/24