Pascal and Current Year

Relevant wiki: Binomial Theorem

By binomial theorem, we have $\displaystyle (1+x)^n = \sum_{k=0}^n \binom nk x^k$ . Putting $x=1$ , $\displaystyle \implies \sum_{k=0}^n \binom nk = 2^n$ . That is the sum of numbers of the $(n+1)$ th row of Pascal triangle is equal to $2^n$ . Therefore, that for the 2018th row is $\boxed{2^{2017}}$ .

Note that:

Row 1: sum of numbers is $2^{1-1}=2^0= 1$

Row 2: sum of numbers is $2^{2-1}=2^1= 2$

Row 3: sum of numbers is $2^{3-1}=2^2= 4$

Row 4: sum of numbers is $2^{4-1}=2^3= 8$

Thus, sum of numbers in the Row 2018 is : $2^{2018-1}= 2^{2017}$