Question

induction question

Answer 1

\[\left(\begin{matrix}a \\ b-1\end{matrix}\right)\left(\begin{matrix}a \\ b\end{matrix}\right)\]

Answer 2

\[\left(\begin{matrix}a \\ b-1\end{matrix}\right) + \left(\begin{matrix}a \\ b\end{matrix}\right)\]**

Answer 3

what is the actual question? show that:\[\left(\begin{matrix}a \\ b-1\end{matrix}\right)+\left(\begin{matrix}a \\ b\end{matrix}\right)=\left(\begin{matrix}a+1 \\ b\end{matrix}\right)\]?

Answer 4

i just need answer, how do u know that it is\[\left(\begin{matrix}a+1 \\ b\end{matrix}\right)\]

Answer 5

it depends on how you want to think about it. There are combinatorial proofs, algebraic proofs, pascals triangle (which i dont consider a proof, but it shows you whats going on).

Answer 6

if you just look at pascals triangle, you get:

1
1 1
1 2 1
1 3 3 1

so on and so forth. The notation:\[\left(\begin{matrix}n \\ k\end{matrix}\right)\] gives you the (k+1)th number on the nth row. each term is also the sum of the two elements above it. that is how you get:\[\left(\begin{matrix}n \\ k-1\end{matrix}\right)+\left(\begin{matrix}n \\ k\end{matrix}\right)=\left(\begin{matrix}n+1 \\ k\end{matrix}\right)\]