Question

Compute the probability of being dealt a four-of-a-kind in poker.

• A. $\frac{{}^{13}{C}_1{}^4{C}_2{}^{12}{C}_1{}^4{C}_1}{{}^{52}{C}_5}$
• B. $\frac{{}^{13}{C}_1{}^4{C}_1{}^{12}{C}_1{}^4{C}_2}{{}^{52}{C}_5}$
• C. $\frac{{}^{13}{C}_1{}^4{C}_3{}^{12}{C}_1{}^4{C}_2}{{}^{52}{C}_5}$
• D. $\frac{{}^{13}{C}_1{}^4{C}_4{}^{12}{C}_1{}^4{C}_1}{{}^{52}{C}_5}$

Answer 1

Answer: (D)

$\frac{{}^{13}{C}_1{}^4{C}_4{}^{12}{C}_1{}^4{C}_1}{{}^{52}{C}_5}$

Four of a kind in $5$ cards.

Four cards should have same number another card will have different number.

Total $13$ numbers.

Selecting one number $={}^{13}{C}_1$.

$4$ cards will have same numbers $={}^4{C}_4$.

From remaining $48$ cards select one card.

Total number of ways$={}^{13}{C}_1\times {}^4{C_4\times }^{12}{C}_1\times {}^4{C}_1$

Total number of ways of selecting $5$ cards $={}^{52}{C}_5$

Total probability$=\frac{{}^{13}{C}_1\times {}^4{C_4\times }^{12}{C}_1{\times }^4{C}_1}{{}^{52}{C}_5}$