What is the probability that the first white ball is seen after the $6$th draw? Hey guys a really easy question that I solved but the solution says otherwise so I need to check if the solution is wrong (hope so).

An urn contains $3$ white balls, $7$ red balls.  Balls are drawn one
  by one without replacement.  What is the probability that the first
  white ball is seen after the 6th draw?

First does this mean the $6$th is white or, after the sixth draw, meaning the seventh?
Anyhow, it means the preceding balls are red so after doing the calculation I keep on getting $\frac 1{40}$. But in the solution (MCQ) it says $\frac 1{30}$
But how can I make a mistake with such a simple question :
MY answer was : $$\frac {7\times6\times5\times4\times3\times3}{10\times9\times8\times7\times6\times5}$$   Took the assumption that there are $5$ reds before, so the sixth is white.
 A: I found the answer , thanks for the commenting for the push , after just calculating Probability (first 6 balls are red) I got the desirable 1/30  which makes sence , since the only constraint is that WE DO NOT OBTAIN WHITE BEFORE THE 6TH. So I guess that was the answer to my problem , tricky words everywhere I look ...
A: 
Let us think of a different situation or Problem. A Plane can be hit by an anti-aircraft gun. The Probability of hitting the plane at the 1st,2nd,3rd,4th,5th, and 6th shots are [P1],[P2][P3],[P4],[P5],[P6] respectively. Then The Probability that the plane is hit in None of the Shots is given by : $[1 -{{P_{1}}}][1 -{{P_{2}}}][1 -{{P_{3}}}][1 -{{P_{4}}}][1 -{{P_{5}}}][1 -{{P_{6}}}]$
Similarly, the probability that the first white ball is seen after the 6th draw means that All these 6 draws are devoid of any White Balls. Therefore we can use the same equation here also as the situation is the same but the words are different. Hence, The Required Probability that the first white ball is seen after the 6th draw = $[1 -{{P_{1}}}][1 -{{P_{2}}}][1 -{{P_{3}}}][1 -{{P_{4}}}][1 -{{P_{5}}}][1 -{{P_{6}}}] = \frac{1}{30}$. Here, The Probability of getting a White ball at the 1st,2nd,3rd,4th,5th, and 6th shots are denoted by $[{{P_{1}}}],[{{P_{2}}}],[{{P_{3}}}],[{{P_{4}}}],[{{P_{5}}}],[{{P_{6}}}]$respectively.
