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I have two questions that I've been working on.

Given: Set A = { a, b, c, d, e, f, g, h }

  1. How many sets of size 5 does A have which contain the element b?

I think I have this one. My solution is 8C5 - 7C5 = 35 sets (Please let me know if this is wrong!).

Here is the second one that throws me off:

    1. How many sets of size 5 does A have which contain the element c but do not contain e?

I am not sure where to start. the "do not contain e" part throws my brain off. Any guidance would be helpful!

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  • $\begingroup$ "the "do not contain e" part throws my brain off" Why? When it comes to picking elements just... don't pick $e$. If you want you can treat it as the $e$ wasn't an element in the set to begin with. How many subsets of $\{a,b,c,d,f,g,h\}$ have five elements and contain $c$. $\endgroup$ – fleablood Oct 15 '19 at 2:39
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1) Another way to do 1) is that you must have $b$ and there are four other elements from the remaining $7$ elements to pick. So there are ${7\choose 4} = 35$.

2) So you must have a $c$ and you must have $4$ other elements but none of them can be $e$. So the other $4$ can be any of $a,b,d,f,g,h$. So there are ${6 \choose 4}$ ways to do this.

Or you can do inclusion exclusion.

${8\choose 5}$ are the total.

${7\choose 5}$ are those that don't contain $c$ so ${8\choose 5} - {7\choose 5}$ are those that do contain $c$.

And ${8\choose 5} - {7\choose 5}$ are those that do contain $b$ so ${8\choose 5} - ({8\choose 5} - {7\choose 5})$ are those that don't contain $b$. (Um... are you being to see we doing things by taking the compliment of things that don't have a property can be .... redundant?)

And then those that do contain $c$ but not $b$ are .... Hmm, okay. I don't know how to do this if compliment of those that do/don't is your only tool.

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A subset of size $5$ that contains $c$ must have $4$ elements other than $c$. There are $7$ elements other than $c$ in $A$, but we're not allowed to choose $e$ so we have only $6$ to choose from. That gives ${6\choose4}=15$. By ${6\choose4}$ I mean the same thing as $_6C_4$.

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