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help plz (repost with LaTeX)

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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
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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
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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