Chapter 9.1, Problem 31E

Solution

To calculate: The numbers of different hands that can be formed if 13-hands include the ace and the king of spades.

The resultant answer is $37,353,738,800$ .

Given information:

In a deck of 52 cards, 13-hands include the ace and the king of spades.

Formula used: The number of combinations of n objects taken r at a time denoted by $C_{n_r}$ and given by $C_{n_r}=\frac{n!}{r!\left(n-r\right)!}$ .

Calculation:

Total number of cards in a deck: 52

Each deck has one spade king and one spade ace. 13-hands include the ace and the king of spades.

Then the number of ways for selecting remaining 11 cards from 50 cards can written as:

  $C_{{50}_{11}}$

Rewrite the expression using the formula $C_{n_r}=\frac{n!}{r!\left(n-r\right)!}$ ,

  $\begin{array}{c}{C}_{{50}_{11}}=\frac{50!}{11!\left(50-11\right)!}\\ =\frac{50!}{11!39!}\end{array}$

Now expand the factorial,

  $\frac{50\cdot 49\cdot 48\cdot 47\cdot 46\cdot 45\cdot 44\cdot 43\cdot 42\cdot 41\cdot 40\cdot 39!}{39!\left(11\cdot 10\cdot 9\cdot 8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1\right)}=37,353,738,800$

Therefore, the value of $C_{{50}_{11}}$ is $37353738800$ .