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# questions

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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
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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"   Apr 6, 2019
#2
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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)   Apr 6, 2019
edited by CPhill  Apr 6, 2019
#3
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3.) What is the least positive integer with exactly 10 factors?

This is given by  :

(2)^4 * 3^1  =   48   Apr 6, 2019