Let $L(n,k)$ be the increasing $k$-tuples from $\{1,\dots,n\}$, listed in lexicographic order.

Eg, for $n=9$, $k=3$, the sequence $L(n,k)$ would be:

$$(1,2,3), (1, 2, 4), (1, 2, 5),\dots,(7, 8, 9).$$

The question is: given that a $k$-tuple $(a_1,\dots,a_k)$ is in position $N$ in $L(n,k)$, with $a_k<n$, is there a formula for the position of $(a_1,\dots,a_k,a_k+1)$ in $L(n,k+1)$ in terms of $N$?

(The original post asked about the case $n=9$, $k=3$, and was a bit difficult to understand, so I have edited it. Please feel free to revert the changes if you wish.)

EDIT [JMP]: Original formula said $n$, but $(7,8,9)\to(7,8,9,10)$ for example.

EDIT [HT]: Rather than change $n$ to $n+1$, I explicitly required $a_k<n$, which seems to have been what the OP intended.