Drawing Ten Cards From Well Shuffled Deck Of 52

#1
What I wanted to know was the probability of drawing one particular card from a standard, well shuffled 52-card deck, when drawing 10 random cards one after another. If all were numbered 1-52, what would be the probability of drawing card #1 showing up in one of the 10 picked. For the first card, it seemed likely to me that the probability would be 1/52 and the second card would be 1/51, the third 1/50 ... and the last 1/43. One-at-a-time, each of those individual probabilities are simple. I wanted the aggregate probability of hitting my target card in a random selection of 10 of the 52. My first instinct was to add them all up, thinking, you know, that the answer would be the sum total of the probabilit(ies) of each of the 10 events. That came up to 21.13%. Took me by surprise and I began to question my logic. Searching the internet for the problem yielded many other sample problems. But mine was nowhere to be found. So my homework consisted of the web search and applying my gut instinct. Then I found you guys, registered 'n now, here I am, asking my question, posing it to a body of experts. Thanks in advance for looking into this and sharing any comments.
 
#3
I'm not, actually. What I wanted to know was whether my 21.13% aggregate probability calculation was correct. I'm guessing the probability of someone unskilled in the art getting the right answer is low.