Gamble - Lottery probability question

Question

A lottery has numbers from 1 to 25. 15 numbers will be randomly drawn, whoever hits the 15 numbers wins the prize. I will bet 17 numbers, 2 more than necessary, to increase my odds. What is the probability that among the 17 numbers I bet I will hit 11 numbers ?

Answer 1

The total number of ways in which $15$ numbers can be drawn is ${25}\choose{15}$. For you to win $11$ numbers first you need to choose $11$ numbers from the $17$ numbers you bet on that is ${17}\choose{11}$ and the rest is ${8}\choose{4}$. So the final answer is $\frac{{{17}\choose{11}} \times {{8}\choose{4}}} {{25}\choose{15}}.$

Answer 2

In other words, what is the probability that we choose $11$ out of $15$ correct numbers and $6$ out of $10$ incorrect numbers when we choose $17$ numbers randomly.

$$ \frac{\binom{15}{11}\times\binom{10}{6}}{\binom{25}{17}} $$