Please Verify

[rgeer]

Hello,

Please Verify the following problem:
If nC6=nC4, then N=
a)2 b)8 c)10 d)24

I get c)10, becaus 10C6=210. and 10C4=210.

Please verify for me.

Thanks in advance,
Ryan

 

[Daniel_Feldman]

Looks good to me!

 

[soroban]

Hello, Ryan!

If \(\displaystyle _nC_6\,=\._nC_4\), then \(\displaystyle N\,=\)

\(\displaystyle a)\;2\qquad\qquad b)\;8\qquad\qquad c)\;10\qquad\qquad d)\;24\)

I get \(\displaystyle c)\;10\), because \(\displaystyle _{_{10}}C_{6}\,=\,210\) and \(\displaystyle _{_{10}}C_{4}\,=\,210\)

Please verify for me.

Your answer is correct . . . but how did you get it?


\(\displaystyle \L_nC_6\): There are \(\displaystyle n\) apples in a bowl; we select \(\displaystyle 6\) of them to remove from the bowl.

\(\displaystyle \L_nC_4\): There are \(\displaystyle n\) apples in a bowl; we select \(\displaystyle 4\) of them to leave in the bowl.

Since these situations are identical, there are \(\displaystyle 10\) applies in the bowl.


If they insist on seeing some algebra . . .

We have: . \(\displaystyle \L _nC_6 \;= \;_nC_4\qquad\Rightarrow\qquad \frac{n!}{6!(n-6)!} \;= \;\frac{n!}{4!(n-4)!}\)

That is: .\(\displaystyle \L4!n!(n-4)! \;= \;6n!(n-6)!\)

. . . \(\displaystyle \L 4!n!(n-4)(n-5)(n-6)! \;= \;6\cdot5\cdot4!n!(n-6)!\)

. . . \(\displaystyle \L (n-4)(n-5) \;= \;30\qquad\Rightarrow\qquad n^2 - 9n + 20 \:= \:30\qquad\Rightarrow\qquad n^2 - 9n - 10 \:= \:0\)

. . . \(\displaystyle \L (n - 10)(n + 1) \:= \:0\qquad\Longrightarrow\qquad n\,=\,10,\,-1\)