Probability distribution table for $5$ ducks and $2$ geese A small farm has $5$ ducks and $2$ geese. Four of these birds are to be chosen at random. The random variable $X$ represents the number of geese chosen.
Draw a probability distribution table for $X$.
I got this far :
$P(X=0) = \frac{1}{7}$
(Duck, duck,duck,duck )
$P(X=1) = \frac{4}{7}$
(Duck, duck, duck, geese;
 Duck, duck, geese, duck;
 Duck, geese, duck, duck;
 Geese, duck, duck, duck) 
$P(X=2) = \frac{6}{7}$
(Duck, duck, geese, geese;
 Duck, geese, duck, geese;
 Geese, duck, geese, duck;
 Geese, geese, duck, duck;
 Geese, duck, duck, geese;
 Duck, geese, geese, duck)
But that would be wrong because addition of all of these should be equal to $1$ (which is not the case here).
What am I doing wrong?
 A: We can use the concept of Combinations here.
For $P (X=0) $, we have that we should select $4$ ducks from the available $5$ as we shouldn't select any geese. So we have the probability as $$P_1 =\frac {\binom {5}{4}}{\binom {7}{4}} $$
For $P (X=1) $, we have that we should select $3$ ducks and $1$ goose to have a total of $4$ selected. So we have the probability as $$P_2=\frac {\binom {5}{3}\cdot \binom {2}{1}}{\binom{7}{4}} $$
Similarly proceed for $P (X=2) $, we can see that $P (X=0) + P (X=1) + P (X=2) $ equals $1$. Hope it helps. 
A: You are doing it wrong. Your case with two geese has wrong answer.
Case 1-  No geese
P(X=0) = $\frac{\binom{5}{4}}{\binom{7}{4}} = \frac{1}{7}$
Case 2-  One goose
P(X=1) = $\frac{\binom{5}{3} \times \binom{2}{1}}{\binom{7}{4}} = \frac{4}{7}$
Case 3-  Two geese
P(X=2) = $\frac{\binom{5}{2} \times \binom{2}{2}}{\binom{7}{4}} = \frac{2}{7}$
Check sum of all probabilities equal to 1.
A: For HN17
Your reasoning for P(x=0) , P(X=1) , and P(x=2) is perfect 
But, the P(X=2) found by you is not right ,because when you find the P(Duck, duck, geese, geese)= 5/7*4/6*2/5*1/4=1/21  which is equal to all the other probs when taken 2G and 2D  , and   a total number of combinations  of taking 2G and 2D   equal to 4C2=6
So, P(X=2)=4C2 *1/21= 6*1/21=2/7 which is  the right value.
Answer given to future readers! 
