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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?

 Apr 6, 2019
 #1
avatar+101150 
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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"

 

 

cool  cool cool

 Apr 6, 2019
 #2
avatar+101150 
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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)

 

 

cool cool cool

 Apr 6, 2019
edited by CPhill  Apr 6, 2019
 #3
avatar+101150 
0

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

 

This is given by  :

 

(2)^4 * 3^1  =   48

 

 

cool cool cool

 Apr 6, 2019

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