Precalculus (6th Edition) Blitzer

a) The required solution is $2598960$ b) The required solution is $1287$ c) The required solution is $\frac{1287}{2598960}$
(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}