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Can someone check these (bit skeptical of my answers).

a) How many copies of $C_4$ in $K_n$?

Picking any 4 vertices can be used to give a copy of $ C_4 $of each of these there are $4!$ ways in which the vertices can be arranged but this overcounts by a factor of 8 (4 vertices $ \times $ 2 directions that can be transversed) so the total is $ (4!/8)* nC4=3(nC_4)$.

b) How many copies of $P_3$ in $K_n$?

Any 4 vertices choose represent $4C3 = 4 $paths so we have 4(nC4) in total.

How can the answer to a be smaller than the answer to b when we have that every$ P_3 $ in $K_n$ is contained within a $C_4$ in $K_n$?

No idea about this part any help?

Thanks!

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  • $\begingroup$ Part a is good, For part b why do you want to use part a? You can solve part b using similar techniques. $\endgroup$
    – Asinomás
    Jan 7 '15 at 21:15
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For the first question look here

For the second question there are $\binom{n}{3}$ ways to select the three vertices in the path. And after that there are $3$ ways to select which of the $3$ vertices is going to be the middle one, so $3\binom{n}{3}$

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