An r-combination of a set of n elements is a subset that contains r of the n elements. The number of r-combinations of a set of n distinct objects is \[C(n,r)=\frac{n!}{r!(n-r)!}\] .
There are C(n,k) ways to select k of the n numbers to be in the bin.
#of possible outcomes=C(n,k)
When we select none of the k winning numbers, then there are C(n-k,k) ways to select the k numbers from the remaining n-k numbers. Note:
\[When\ n-k, then\ C(n-k,k)=0\]
#of favorable outcomes=C(n-k,k)
The probability is the number of favorable outcomes divided by the number of possible outcomes:
\[P(no\ winnning\ numbers)=\frac{\#of\ favorable\ outcomes}{\#of\ possible\ outcomes} \\
=\frac{C(n-k,k)}{C(n,k)} \\
=\frac{(n-k)!/k!(n-2k)!)}{n!/k!(n-k)!} \\
=\frac{[(n-k)!]^{2}}{n!(n-2k)!} \\\]
The sum of the probability of an event and the probability of the complementary ebent needs to be equal to 1 \[(P(E)+P(\overline{E})=1)\], which implies that the probability of the complementary enevt os 1 decreased by the probability of the event
\[P(at\ least\ one\ winning\ number)=1-P(no\ winning\ number) \\
1-\frac{[(n-k)!]^{2}}{n!(n-2k)!}\]