For $^nC_k$, where $n = 15$ and $k = 1 \to 15$, is there a formula that finds this total without using summation?
A shorter example that uses summation:
$$\begin{align} ^{15}C_1 &= 15 \\ ^{15}C_2 &= 105 \\ \implies {^{15}C_1} + {^{15}C_2} &= 120 \end{align}$$
I feel like there should be a modification (e.g. $n+1$ or something) that allows the summation to be found without adding up all the separate solutions for $k = 1 \to 15$.
Thanks!
As Frpzzd said, you can look at the binomial theorem and related topics.
The relevant formula you're looking for is:
$$\sum_{k = 0}^{n} \binom{n}{k} = 2^n$$
Since you're only summing from $1$ to $n$ your result would be $2^n -1$
Source for further reading: https://brilliant.org/wiki/binomial-theorem-n-choose-k/#pascals-triangle