Find the sum of $\binom{100}1 + 2\binom{100}2 + 4\binom{100}3 +8\binom{100}4+\dots+2^{99}\binom{100}{100}$ Find the sum of 
$\binom{100}1 + 2\binom{100}2 + 4\binom{100}3 +8\binom{100}4+\dots+2^{99}\binom{100}{100}$
How you guys work on with this question? With the geometric progression? Combination? Or anyother way to calculate?

$$\sum_{r=1}^{100}2^{r-1}\binom{100}r=\frac12\sum_{r=1}^{100}2^r\binom{100}r=\frac12\left[(1+2)^{100}-1\right]$$
