Chapter 9, Problem 11RE

Solution

To calculate: The numbers of different 13-card bridge hands that includes the ace, king, and queen of spades.

The resultant answer is $8,217,822,536$ .

Given information:

The given statement says to find out numbers of different 13-card bridge hands that include includes the ace, king, and queen 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

13 cards are to be selected which includes include the ace, king, and queen of spades.

Now note that out of 13 cards 3 cards have already been selected. Now 10 cards have to be selected from the deck.

Then the remaining card in the deck will become: $52-3=49$

Now from these 49 cards, 10 cards are to be selected: $C_{{49}_{10}}$

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

  $\begin{array}{c}{C}_{{49}_{10}}=\frac{49!}{10!\left(49-10\right)!}\\ =\frac{49!}{10!39!}\end{array}$

Now expand the factorial,

  $\frac{49\cdot 48\cdot 47\cdot 46\cdot 45\cdot 44\cdot 43\cdot 42\cdot 41\cdot 40\cdot 39!}{39!\left(10\cdot 9\cdot 8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1\right)}=8,217,822,536$

Therefore, the total number of ways of selecting 13-card bridge hands that include includes the ace, king, and queen of spades is $8,217,822,536$ .