# A pack of 36 cards includes 20 numbered cards from 6 to 10 inclusive, 4 aces and 12 picture cards. If a hand of 5 cards is selected at random, how do you find the probability of receiving at least 2 aces?

31,776

#### Explanation:

Let's first see that if I do this, I can say:

Hands with 0 Aces + Hands with 1 Ace + Hands with 2 Aces + Hands with 3 Aces + Hands with 4 Aces = All possible hands

And so we can work this by adding up the hands with 2, 3, and 4 aces, or we can find all possible hands and subtract out hands with 0 and 1 ace. Since they're the same amount of work, I'll do the subtraction method (it being more interesting).

The number of All possible hands is combination of 36 pick 5:

((36),(5))=(36!)/((31!)(5!))=(36xx35xx34xx33xx32)/120="376,992"

The number of hands with 0 aces means we have 0 aces from the 4 available and we have 5 cards from the 32 remaining:

$\left(\begin{matrix}4 \\ 0\end{matrix}\right) \left(\begin{matrix}32 \\ 5\end{matrix}\right) = \left(1\right) \frac{32 \times 31 \times 30 \times 29 \times 28}{120} = \text{201,376}$

And the number of hands with 1 ace means we have 1 ace from the 4 available and 4 cards from the remaining 32:

$\left(\begin{matrix}4 \\ 1\end{matrix}\right) \left(\begin{matrix}32 \\ 4\end{matrix}\right) = \left(4\right) \frac{32 \times 31 \times 30 \times 29}{24} = \text{143,840}$

This gives:

$\text{376,992"-"201,376"-"143,840"="31,776}$

~~~~~

Since this might seem like a very small number compared to what we've found so far, let's work out the hands with 2, 3, and 4 aces:

2 aces:

$\left(\begin{matrix}4 \\ 2\end{matrix}\right) \left(\begin{matrix}32 \\ 3\end{matrix}\right) = \left(6\right) \frac{32 \times 31 \times 30}{6} = \text{29,760}$

3 aces:

$\left(\begin{matrix}4 \\ 3\end{matrix}\right) \left(\begin{matrix}32 \\ 2\end{matrix}\right) = \left(4\right) \frac{32 \times 31}{2} = 1984$

4 aces:

$\left(\begin{matrix}4 \\ 4\end{matrix}\right) \left(\begin{matrix}32 \\ 1\end{matrix}\right) = \left(1\right) \left(32\right) = 32$

$\text{29,760"+1984+32="31,776}$