Question

What is the number of ways of choosing $4$ cards from a pack of $52$ playing cards? In how many of these (i) four cards are of the same suit (ii) are face cards (iii) two are red cards and two are black cards.

Answer 1

Number of ways of choosing $4$ cards from $52$ cards =${}^{52}{C}_4$

1) Four cards of same suit, they are diamond, spade, heart, club.

Each suit contains $13$ cards
$\Rightarrow$number of ways to select 4 cards of same suit in each suit =${}^{13}{C}_4$
Total number of ways to four cards of same suit ${=}^{13}{C}_4{\times }^{13}{C}_4{\times }^{13}{C}_4{\times }^{13}{C}_4$

$=4{\times }^{13}{C}_4$
$=4\times \frac{13\times 12\times 11\times 10}{4\times 3\times 2\times \times 1}$
$=2860$ ways

2) Face cards :

There are $12$ face cards in a deck. In each suit there are $3$ face cards, queen, king and Jack.

Number of ways of choosing $4$ cards from $12$ face cards =${}^{12}{C}_4$

$=\frac{12\times 11\times 10\times 9}{1\times 2\times 3\times 4}$
$=495$ ways.

3) $2$ are red and $2$ are black

There are $26$ red and $26$ black cards in a deck.

Number of ways of choosing $2$ red and $2$ black cards from $52$ cards =${}^{26}{C}_2{\times }^{26}{C}_2={(}^{26}{C}_2{)}^2$
$=(\frac{26\times 25}{1\times 2}{)}^2$
$=(325{)}^2$
$=105625$