Initially I thought it would be
$\left(\frac{1}{70}\right)^5\left(\frac{1}{25}\right)$
to get the probability of choosing the right number out of 70 exactly 5 times and then the 6th number out of 25.
Then I saw that it was actually
${70\choose{5}}*25$
What exactly is the difference and why is the first one not correct?
For your attempts, you wouldn't want to repeat the same number. Once a number is chosen, you wouldn't want to choose it again. Also order shouldn't matter.
Also, for the answer that you show us, that can't be a probability, as it is the number of possibility. Just invert it to get the probability.