On a lottery ticket, you need to mark 8 squares out of 64. What is the probability that after the draw, in which 8 of the 64 squares will also be chosen (all such possibilities are equally likely), it will turn out that
a) exactly 4 squares are guessed?
b) exactly 5 squares are guessed?
c) all 8 squares are guessed?
8 cells out of 64 can be chosen in $C_{64}^{8}$ ways. Therefore, the probability that a specific combination will occur is $\frac{1}{C_{64}^{8}}$. The number of cases in which exactly $k$ cells are guessed correctly is equal to the number of ways to choose $k$ cells from the 8 marked and $8-k$ cells from the 56 unmarked, which is $C_{8}^{k} \cdot C_{56}^{8-k}$. Thus, the probability of guessing exactly $k$ cells correctly is $\frac{C_{8}^{k} \cdot C_{56}^{8-k}}{C_{64}^{8}}$.

## Answer

a) $\frac{C_{8}^{4} \cdot C_{56}^{4}}{C_{64}^{8}}$
b) $\frac{C_{8}^{5} \cdot C_{56}^{3}}{C_{64}^{8}}$
c) $\frac{1}{C_{64}^{8}}$.

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