1.) In the land of Noom, all nouns are 4-letter words with consonants at the beginning and end and a repeated vowel (a, e, i, o, or u) in the middle. How many such words are possible?

2.) How many positive three-digit integers are there in which the sum of its three digits is odd?

3.) What is the least positive integer with exactly 10 factors?

Guest Apr 6, 2019

#1**0 **

1.) In the land of Noom, all nouns are 4-letter words with consonants at the beginning and end and a repeated vowel (a, e, i, o, or u) in the middle. How many such words are possible?

We have 21 ways of choosing the first consonant....5 ways of choosing a repeated vowel and (assuming that consonants can also be repeated), 21 ways of choosing the last consonant

So 21 * 5 * 21 = 2205 "nouns"

CPhill Apr 6, 2019

#2**0 **

2.) How many positive three-digit integers are there in which the sum of its three digits is odd?

We have the following possibilities :

2 integers are even and 1 is odd or all three are odd

First case

(odd)(even)(even) = (5)(5)(5) = 125

(even)(odd)(even) = (4)(5)(5) = 100

(even)(even)(odd) = (4)(5)(5) = 100

Total = 325

Second case

(odd)(odd)(odd) = (5) (5) (5) = 125

So...the total is 325 + 125 = 450

(Which is what we might expect...since there are 900 3-digit numbers....and half of them should have odd sums of their integers)

CPhill Apr 6, 2019