Question

plse solve this sum

Answer 1

The total number of distinct hands is simply 52 choose 5
\[ {52\choose 5} = 2,598,960 \]
The total number of distinct hands with NO kings is
\[ {48\choose 5} = 1,712,304 \]
The total number of distinct hands with at least one king is the difference,
\[ {52 \choose5} - {48 \choose 5} = 886,656 \]

Answer 2

Alternatively, I could find the probability that a hand of five cards has no kings:
\[P(\text{No Kings}) = \frac{48}{52}\cdot \frac{47}{51} \cdot \frac{46}{50} \cdot \frac{45}{49} \cdot \frac{44}{48} = \frac{35673}{54145}\]
So
\[P(\text{ At Least One King} ) = \frac{18472}{54145} = \frac{\text{Hands with at least 1 King}}{\text{Total Hands}} \]
Multiplying that probability by 52 choose 5, the total number of hands, yields the same result.