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Suppose we define \(\ell(n)\) as follows: If \(n\) is an integer from \(0\) to \(20,\) inclusive, then \(\ell(n)\) is the number of letters in the English spelling of the number \(n;\) otherwise, \(\ell(n)\) is undefined. For example, \(\ell(11)=6\) because "eleven" has six letters, but \(\ell(23)\) is undefined, because \(23\) is not an integer from \(0\) to \(20\)

How many numbers are in the domain of \(\ell(n)\) but not in the range of \(\ell(n)\)?

 

Suppose we define $\ell(n)$ as follows: If $n$ is an integer from $0$ to $20,$ inclusive, then $\ell(n)$ is the number of letters in the English spelling of the number $n;$ otherwise, $\ell(n)$ is undefined. For example, $\ell(11)=6,$ because "eleven" has six letters, but $\ell(23)$ is undefined, because $23$ is not an integer from $0$ to $20.$

How many numbers are in the domain of $\ell(n)$ but not the range of $\ell(n)?$

 Mar 18, 2019
 #1
avatar+4772 
+1

There's no comprehensive formula for the number of letters in the English spelling of a given number.

So you've got to just list them all out and check.

 

\(\text{The range of }\ell(n) \text{ is }(3,4,\dots,8)\\ \text{Thus }|(1,2) \cup (9,10,\dots, 20)| = 14\\ \text{is the number you are after}\)

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 Mar 18, 2019
edited by Rom  Mar 18, 2019
 #2
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Hmmm...That was incorrect...

Guest Mar 18, 2019
 #3
avatar+4772 
+1

Ok, I spelled eighteen wrong.

 

9 isn't in the range either.  So 14 are in the domain and not in the range.

 

Sorry for the confusion.

Rom  Mar 18, 2019
 #4
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Okay thanks.  That was right!

Guest Mar 19, 2019

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