Chapter 13.1, Problem 45E

a.

To determine

To calculate: The number of 5 card hands are possible that have 3 hearts and 2 clubs from a deck of 52 cards.

Answer to Problem 45E

The number of 5 card hands are possible that have 3 hearts and 2 clubs from a deck of 52 cardsare $22308$ .

Explanation of Solution

Given information:

There is deck of 52 cards. 5 cards combinations must be there with 3 hearts and 2 clubs.

Formula used:

If there are n objects taken r at a time then combination is defined as $C\left(n,r\right)=\frac{n!}{r!\left(n-r\right)!}$ .

Calculation:

Consider the provided information that there is deck of 52 cards. 5 cards combinations must be there with 3 hearts and 2 clubs.

In a deck of 52 cards there 13 hearts and 13 clubs.

3 cards of heart and 2 of clubs are to be selected, use combination as selection does not matter.

The possible ways are $C\left(13,3\right)\cdot C\left(13,2\right)$ .

Recall that if there are n objects taken r at a time then combination is defined as $C\left(n,r\right)=\frac{n!}{r!\left(n-r\right)!}$ .

Here n is 13and r is 3 for $C\left(13,3\right)$ and n is 13and r is 2for $C\left(13,2\right)$ .

  $\begin{array}{c}C\left(13,3\right)\cdot C\left(13,2\right) =\frac{13!}{3!\left(13-3\right)!}\cdot \frac{13!}{2!\left(13-2\right)!}\\ =\frac{13\cdot 12\cdot 11\cdot 10!}{3!\cdot 10!}\cdot \frac{13\cdot 12\cdot 11!}{2!\cdot 11!}\\ =13\cdot 2\cdot 11\cdot 13\cdot 6\\ =22308\end{array}$

Thus, the number of 5 card hands are possible that have 3 hearts and 2 clubs from a deck of 52 cards are $22308$ .

b.

To determine

To calculate: The number of 5 card hands are possible that have 1 ace, 2 jacks and 2 kings from a deck of 52 cards.

Answer to Problem 45E

The number of 5 card hands are possible that have 1 ace, 2 jacks and 2 kings from a deck of 52 cards are $144$ .

Explanation of Solution

Given information:

There is deck of 52 cards. 5 cards combinations must be there with 1 ace, 2 jacks and 2 kings.

Formula used:

If there are n objects taken r at a time then combination is defined as $C\left(n,r\right)=\frac{n!}{r!\left(n-r\right)!}$ .

Calculation:

Consider the provided information that there is deck of 52 cards. 5 cards combinations must be there with 1 ace, 2 jacks and 2 kings.

In a deck of 52 cards there 4 aces, 4 jacks and 4 kings.

1 ace, 2 jacks and 2 kings are to be selected, use combination as selection does not matter.

The possible ways are $C\left(4,1\right)\cdot C\left(4,2\right)\cdot C\left(4,2\right)$ .

Recall that if there are n objects taken r at a time then combination is defined as $C\left(n,r\right)=\frac{n!}{r!\left(n-r\right)!}$ .

Here n is 4 and r is 1 for $C\left(4,1\right)$ and n is 4 and r is 2for $C\left(4,2\right)$ .

  $\begin{array}{c}C\left(4,1\right)\cdot C\left(4,2\right)\cdot C\left(4,2\right) =\frac{4!}{1!\left(4-1\right)!}\cdot \frac{4!}{2!\left(4-2\right)!}\cdot \frac{4!}{2!\left(2\right)!}\\ =\frac{4\cdot 3!}{3!}\cdot \frac{4\cdot 3\cdot 2!}{2!\cdot 2!}\cdot \frac{4\cdot 3\cdot 2!}{2!\cdot 2!}\\ =4\cdot 6\cdot 6\\ =144\end{array}$

Thus, the number of 5 card hands are possible that have 1 ace, 2 jacks and 2 kings from a deck of 52 cards are $144$ .

c.

To determine

To calculate: The number of 5 card hands are possible that have all face cards from a deck of 52 cards.

Answer to Problem 45E

The number of 5 card hands are possible that have all face cardsfrom a deck of 52 cards are $792$ .

Explanation of Solution

Given information:

There is deck of 52 cards. 5 cards combinations must be there with all face cards.

Formula used:

If there are n objects taken r at a time then combination is defined as $C\left(n,r\right)=\frac{n!}{r!\left(n-r\right)!}$ .

Calculation:

Consider the provided information that there is deck of 52 cards. 5 cards combinations must be there with all face cards.

In a deck of 52 cards there 12 face cards.

All face cardsare to be selected, use combination as selection does not matter.

The possible ways are $C\left(12,5\right)$ .

Recall that if there are n objects taken r at a time then combination is defined as $C\left(n,r\right)=\frac{n!}{r!\left(n-r\right)!}$ .

Here n is 12 and r is 5for $C\left(12,5\right)$ ,

  $\begin{array}{c}C\left(12,5\right) =\frac{12!}{5!\left(12-5\right)!}\\ =\frac{12\cdot 11\cdot 10\cdot 9\cdot 8\cdot 7!}{5!\cdot 7!}\\ =\frac{12\cdot 11\cdot 10\cdot 9\cdot 8}{120}\\ =792\end{array}$

Thus, the number of 5 card hands are possible that have all face cardsfrom a deck of 52 cards are $792$ .