Question

Determine the number of $5$ card combination out of a deck of $52$ cards if each selection of $5$ cards has at least one king.

Answer 1

The observation that in a deck of $52$ cards we have $4$ kings and $48$ non kings.

Selection of $5$ cards having at least one king can be made as follows:

$1$ king and $4$ non kings or $2$ kings and $3$ non kings or $3$ kings and $2$ non kings or $4$ kings and $1$ non king.

This can be expressed mathematically as

Number of selections: ${}^4{C}_1{\times }^{48}{C}_4{+}^4{C}_2{\times }^{48}{C}_3{+}^4{C}_3{\times }^{48}{C}_2{+}^4{C}_4{\times }^{48}{C}_1$

$=\frac{4!}{3!1!}\times \frac{48!}{44!4!}+\frac{4!}{2!2!}\times \frac{48!}{45!3!}+\frac{4!}{3!1!}\times \frac{48!}{46!2!}+\frac{48!}{47!1!}$

$=4\times \frac{48\times 47\times 46\times 45}{4\times 3\times 2}+\frac{48\times 47\times 46}{3\times 2}+4\times \frac{48\times 47}{2}+48$

$=778320+103776+4512+48$

$=886656$