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"Given a set $\{1,\ 2,\ 3,\ 4\}$, how many sequences with a length of $4$ with entries from this set have exactly one entry equal to $1$?"

Here is my work so far:
$X = \left\{\text{sequences with length 4 from}\ \{1, 2, 3, 4\}\ \text{with exactly one entry equal to $1$}\right\}$ $Y = \left\{\text{permutations of length 4 from}\ \{1, 2, 3, 4\}\right\}$

By definition, $|Y| = 4\times 4\times 4\times4 = 256$.

Where do I go from here? Should I find $|X|$ as well? I think (by intuition, I haven't checked) that I need to find $|X \cap Y|$ in order to solve this problem. Is this true?

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    Oh, sorry. I'm new to this, so the terminology is also new to me. What you said was what I meant, actually.2012-10-10

2 Answers 2

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Let’s straighten out your terminology first: permutations of $\{1,2,3,4\}$ are automatically of length $4$, and there are $4!$ of them, not $4^4$; there are $4^4$ $4$-term sequences of elements of the set $\{1,2,3,4\}$.

You want nothing to do with permutations of $\{1,2,3,4\}$, and you certainly need to find $|X|$: that’s what the question asks for! To do this, note that each $4$-term sequence in $X$ can be constructed by deciding first which term of the sequence is to be the $1$ and then what members of $\{2,3,4\}$ are to be assigned to the other terms of the sequence. There are $4$ ways to choose the term that’s to be $1$. Once that’s settled, each of the other terms can be given any of $3$ values, $2,3$, or $4$, so it takes $3$ $3$-way choices to pin down the rest of the sequence. Thus, we end up with $4\cdot3\cdot3\cdot3=108$ possible sequences.

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    @user41419: You’re welcome! Elementary combinatorics problems range from quite easy, like this one, to fiendishly difficult, but if you’re just starting, you’ll probably find that most of them will succumb fairly readily if you break them down into things that you can count easily, as I did here.2012-10-10
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The $1$ can be in any one of $4$ places. For each placement of the $1$, there are $3^3$ ways to fill in the rest of the entries.

Note that the problem did not say that the other entries are distinct. It said only that they are not $1$. So for example $4143$ is one of our sequences.