Question

Determine the number of 5 cards combinations out of a deck of 52 cards if at least one of the 5 cards has to be a king ?

Answer 1

Out of the 52 cards 4 are kings and 48 are Non-kings.

Five cards with at least one king

= (one king and 4 non-kings) or (two kings and 3 non kings) or (3 kings and 2 non kings) or (4 kings and 1 non kings)

$={(}^4{C}_1{\times }^{48}{C}_4)+{(}^4{C}_2{\times }^{48}{C}_3)+{(}^4{C}_3{\times }^{48}{C}_2)+{(}^4{C}_4{\times }^{48}{C}_1)=4\times \frac{48\times 47\times 46\times 45}{4\times 3\times 2}+\frac{4\times 3}{2}\times \frac{48\times 47\times 46}{3\times 2}+4\times \frac{48\times 47}{2}+1\times 48=778320+103776+4512+48=886656$

Required Number of ways = 886656