Find the number of 5-card combinations out of a deck of 52 cards if a least
Question:

Find the number of 5-card combinations out of a deck of 52 cards if a least one of the five cards has to be king.

 

Solution:

Since there are 52 cards in a deck out of which 4 are king and others are nonkings.

So, the no. of ways are as follows:

1. 1 king and 4 non-king

2. 2 king and 3 non-king

3. 3 king and 2 non-king

4. 4 king and 1 non-king

So, total no. of ways are

$={ }^{4} C_{1} \times{ }^{48} C_{4}+{ }^{4} C_{2} \times{ }^{48} C_{3}+{ }^{4} C_{3} \times{ }^{48} C_{2}+{ }^{4} C_{4} \times{ }^{48} C_{1}$

Applying ${ }^{n} C_{r}=\frac{n !}{r !(n-r) !}$

= 778320+103776+4512+48

= 886656 ways.

 

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