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 $\ell(n)?$

Guest Mar 18, 2019

#1**+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.

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

.Rom Mar 18, 2019