Question

Two cards are drawn from a well shuffled pack of $52$ cards. Find the probability that one picked up card is a king card and other card is not a king card.

• A. $\frac{{}^4{C}_1\times {}^{48}{C}_1}{{}^{52}{C}_2}$
  
• B. $\frac{{}^{48}{C}_2}{{}^{52}{C}_2}$
  
• C. $\frac{{}^4{C}_2}{{}^{52}{C}_2}$
  
• D. $\frac{{}^4{C}_2}{{}^{48}{C}_2}$
  

Answer 1

The correct option is A $\frac{{}^4{C}_1\times {}^{48}{C}_1}{{}^{52}{C}_2}$


There are $4$ kings available in a standard deck of cards.
$2$ cards can be selected from $52$ cards in ${}^{52}{C}_2$ ways.

Selection of $1$ king from $4$ kings can be done in ${}^4{C}_1$ ways.

We need to select one non-king card. There would be $52-4=48$ non-king cards.

Selection of $1$ non-king card from $48$ non-king cards can be done in ${}^{48}{C}_1$ ways.

Let $\text{P(E)}$ denotes the probability of drawing one king card and one non-king card.

Number of favorable outcomes $=$ Selection of one king card and one non-king card $={}^4{C}_1\times {}^{48}{C}_1$

Total number of outcomes $=$ Selection of two cards from the deck $={}^{52}{C}_2$

$\text{P(E)}=\frac{\text{Selection of one king card and one non-king card}}{\text{Selection of two cards from the deck}}$

$\text{P(E)}=\frac{{}^4{C}_1\times {}^{48}{C}_1}{{}^{52}{C}_2}$

Therefore, option (a.) is the correct answer.