Question

Permutation/Combination Derek shuffled a pack of 52 playing cards and asked his friends, Ian to choose any 5 cards. How many different 5 cars hands are possible?

Answer 1

SOLUTION

Given the question in the question tab, the following are the solution steps to get possible 5 card hands

Step 1: Identify the type of problem in the question

Ian is choosing a subset of size 5 from a set of size 52. Note that the order in which he is dealt these 5 cards is irrelevant. Thus, order doesn’t matter. Thus, this is a combination problem.

Step 2: Write the formula for finding combination

$\text{nCr}=\frac{n!}{(n-r)!r!}$

Step 3: Find the possible different 5 card hands

$\begin{gathered} n=52,r=5 \\ \text{nCr}=\frac{n!}{(n-r)!r!} \\ 52\text{C5}=\frac{52!}{(52-5)!5!}=\frac{52!}{47!5!} \\ =\frac{52\times51\times50\times49\times48\times47!}{47!\times5!} \\ =\frac{52\times51\times50\times49\times48}{5!}=2598960 \end{gathered}$

Hence, there are 2598960 different possible card hands.