Probability- Gravner -17.c

Question

A bag contains $80$ balls numbered {1, . . . , 80}. Before the game starts, you choose $10$ different numbers from among {1, . . . , 80} and write them on a piece of paper. Then $20$ balls are selected (without replacement) out of the bag at random.

(c) What is the probability that exactly $4$ of your numbers are selected?

Answer 1

Total no. of possible  outcomes are $\binom{80}{20}$

Since exactly 4 balls got matched with the numbers. therefore 4 balls are from the 10 numbers and remaining 16 balls (out of 20) are from the remaining 70 numbers.

Therefore the probability that exactly 4 of the numbers are selected  = $\frac{\binom{10}{4} \times \binom{70}{16}}{\binom{80}{20}}$