Colored Balls

A bowl contains $50$ colored balls: $13$ green, $10$ red, $9$ blue, $8$ yellow, $6$ black, and $4$ white. If you are blindfolded, what is the smallest number of balls you must pick to guarantee that you have at least $7$ balls of the same color?
Source: NCTM Mathematics Teacher 2006

SOLUTION
It is like winning the lottery if you can pick out $7$ balls of the same color one right after another. More likely you will pick the following balls (not in any particular order)
$6$ black (don’t want)
$4$ white (don’t want)
$6$ green ($1$ shy of $7$)
$6$ red ($1$ shy of $7$)
$6$ blue ($1$ shy of $7$)
$6$ yellow ($1$ shy of $7$)
Total = $34$ balls
If you then pick one more ball (doesn’t matter what color), you will have $7$ balls of the same color. $35$ is the smallest number of balls you must pick to guarantee that you have at least $7$ balls of the same color.

Answer: $35$