Question

in how many ways can we draw 3 different cards from a deck of 52 cards, where the order in which cards are drawn doesn’t matter?

Answer 1

$\huge \sf \underline{Concept :}$

Permutations are for lists (order matters)

Combinations are for groups (order doesn't matter)

$\huge \sf \underline{ Explaination :}$

3 different cards can be selected from 52 cards in $\sf \fbox{{}^{52} C_{3} }$ ways

Thus ,

$\mapsto \sf Total \: number \: of \: ways = {}^{52} C_{3} \\ \\ \mapsto \sf Total \: number \: of \: ways = \frac{52!}{3!(52 - 3)!} \\ \\ \mapsto \sf Total \: number \: of \: ways = \frac{52!}{3!(49)!} \\ \\ \mapsto \sf Total \: number \: of \: ways = \frac{52 \times 51 \times \times 50 \times \cancel{49 !}}{3 \times 2 \times 1 \times \cancel{49! }} \\ \\ \mapsto \sf Total \: number \: of \: ways = \frac{132600}{6} \\ \\ \mapsto \sf Total \: number \: of \: ways = 22100 \: \: ways$

Hence , There are 22100 ways to draw 3 different cards from a deck of 52 cards where order doesn't matter