Question

@ganeshie8 Need help with Combinations

Answer 1

If \[\left(\begin{matrix}n \\ 12\end{matrix}\right)=\left(\begin{matrix}n \\ 9\end{matrix}\right)\] find \[\left(\begin{matrix}n \\ 17\end{matrix}\right)\] and \[\left(\begin{matrix}22 \\ n\end{matrix}\right)\]

Answer 2

\[*\left(\begin{matrix}n \\ 12\end{matrix}\right)=\left(\begin{matrix}n \\ 8\end{matrix}\right)\]

Answer 3

Do you know about the symmetry of the binomial coefficient/combination formula?

Answer 4

uh no

Answer 5

maybe be i know but i dont know what its called

Answer 6

It's easily seen in the Pascal triangle:
\[\begin{array}{c|cccccc}
n\backslash k&0&1&2&3&4&\cdots\\
\hline
0&1\\
1&1&1\\
2&1&2&1\\
3&1&3&3&1\\
4&1&4&6&4&1\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots
\end{array}\]

Answer 7

If it's not clear, the the value in the table is \(\dbinom nk\), the row an the top indicates \(n\) and the column to the left is \(k\).

Answer 8

oh okay

Answer 9

For example, when \(n=3\), you get \(\dbinom 31=\dbinom 32=3\). This is what I mean by symmetry.

Answer 10

oh okay yea i know that pattern The numbers on the left side have identical matching numbers on the right side, like a mirror image.

Answer 11

Right. So given that \(\dbinom n{12}=\dbinom n9\), you have enough info to determine the value of \(n\).

Answer 12

my bad that should (n8) not (n9)

Answer 13

\[\left(\begin{matrix}n \\ 12\end{matrix}\right)=\left(\begin{matrix}n \\ 8\end{matrix}\right)\]

Answer 14

Right, I just saw that... Anyway, knowing that \(\dbinom n{12}=\dbinom n8\), we then know that \(\dbinom n{11}=\dbinom n{9}\), then the middle number of that row is given by \(\dbinom n{10}\) (the bottom numbers close in on a midpoint, so to speak).

Answer 15

ok, yea

Answer 16

From the table you should be able to see that when \(n\) is even, you get that one center value that's not repeated. For example, with \(n=2\), that center value is \(\dbinom 21=2\), and for \(n=4\) the center is \(\dbinom 42=6\). In general, then, if \(n=2k\) (i.e. \(n\) is even), we can say that \(\dbinom{2k}k\) is the center value.

For this problem, you have the center \(\dbinom n{10}=\dbinom{2k}k\) which means \(k=10\), and so \(n=2k=20\).

Answer 17

To check: http://www.wolframalpha.com/input/?i=20+choose+12+%3D+20+choose+8

Answer 18

It was that simple?

Answer 19

Yep, it's mostly pattern recognition. Don't forget to evaluate the other coefficients.

Answer 20

wait how do you know 10 was a midpoint?

Answer 21

That conclusion was based on the pattern for other even values of \(n\).

Answer 22

okay, I understand on how to this now. Thanks :D

Answer 23

yw