Answer
a)
The required solution is $2598960$
b)
The required solution is $1287$
c) The required solution is $\frac{1287}{2598960}$
Work Step by Step
(a)
We have to calculate the total number of possible five card poker hands. We can choose 5 cards from the deck of 52 cards.
$\begin{align}
& {}^{52}{{C}_{5}}=\frac{52!}{5!\left( 52-2 \right)!} \\
& =\frac{52\cdot 51\cdot 50\cdot 49\cdot 48\cdot 47!}{5!47!} \\
& =\frac{52\cdot 51\cdot 50\cdot 49\cdot 48}{5\cdot 4\cdot 3\cdot 2\cdot 1} \\
& =2598960
\end{align}$
(b)
We have to calculate the possible number of diamond flushes. We choose 5 cards from 13 cards.
$\begin{align}
& {}^{13}{{C}_{5}}=\frac{13!}{(13-5)!5!} \\
& =\frac{13\cdot 12\cdot 11\cdot 10\cdot 9\cdot 8!}{8!5!} \\
& =\frac{13\cdot 12\cdot 11\cdot 10\cdot 9}{5\cdot 4\cdot 3\cdot 2\cdot 1} \\
& =1287
\end{align}$
(c)
We know that the probability of being dealt a diamond flush is:
$\begin{align}
& P\left( E \right)=\frac{\text{number of}\ \text{possible}\ \text{outcomes}}{\text{total}\ \text{outcomes}} \\
& =\frac{1287}{2598960}
\end{align}$
