1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
allochka39001 [22]
3 years ago
6

Find the number of possible 5-card hands that contain the cards specified from a standard 52-card deck.

Mathematics
1 answer:
Butoxors [25]3 years ago
4 0
A. A "red" card has either the heart or diamond suit. There are 26 of these. If we don't care about the order in which the cards are drawn (which is the assumption I'm using throughout this answer), then there are

\dbinom{26}5=\dfrac{26!}{5!(26-5)!}=65,780

such hands. To be clearer about the fact that we don't want "black" cards, we could also have written

\dbinom{26}5\dbinom{26}0

That is, we want 5 red cards but 0 black cards in our hand; there are 26 of each color to choose from. But the second term reduces to 1 and makes no difference anyway.

b. Now we want 4 spades (there are 13 in the deck) and 1 non-spade (39 non-spades in the deck) so we can make

\dbinom{13}4\dbinom{39}1=27,885

such hands.

c. Now we want 3 face cards (there are 3 per suit, with 4 suits altogether, giving a total of 12 face cards in the deck) and 2 non-face cards (40 in total), so we have

\dbinom{12}3\dbinom{40}2=171,600

possible hands.

d. Now we want 2 aces (4 total in the deck) and 3 non-aces (48 remaining), so we have

\dbinom42\dbinom{48}3=103,776

e. Now, we want at most 1 diamond. So either there are no diamonds or exactly 1 diamond in the hand. There are

\dbinom{13}0\dbinom{39}5+\dbinom{13}1\dbinom{39}4=1,645,020

such possible hands.

f. Finally, we want at least 1 king in the hand, which means we can have between 1 and 4 kings. We can add up the possible hands like in part (e),

\dbinom41\dbinom{48}4+\dbinom42\dbinom{48}3+\dbinom43\dbinom{48}2+\dbinom44\dbinom{48}1

but computing that can get tedious very quickly.

Instead, let's consider the number of hands *not* containing any kings at all. There are

\dbinom40\dbinom{48}5=1,712,304

such possible hands.

With the event of getting at least 1 king in mind, we can see that every possible 5 card hand either has at least 1 king or it does not. These events are disjoint, so

\text{total 5 card hands}=\text{total hands with at least 1 king}+\text{total hands without kings}

So the number of at-least-1-king hands is

\dbinom{52}5-\dbinom40\dbinom{48}5=886,656
You might be interested in
How to make 1.4 into a fraction
inessss [21]

As an improper fraction, the answer is 7/5.

As a mixed number, the answer is 1 2/5.

4 0
3 years ago
Read 2 more answers
How many inches longer is a hercules beetle than a ladybug
Mila [183]

Answer:

A Hercules Beetle is 6.97 inches larger than A ladybug

Step-by-step explanation:

A hercules beetle is 7 inches long while a ladybug is about 0.03 cm long when subtracting these two you will get a grand total of 6.97.

6 0
2 years ago
kiyo bought a pizza for 12.75 and four medium drinks to Pauli’s pizza. Define a variable and write an expression to represent th
Stels [109]

The total cost, if one drink costs $3 is $27.75

  • Let the cost of a pizza be x

  • Let the cost of a drink be y

If kiyo bought a pizza for 12.75 and four medium drinks to Pauli’s pizza, this expressed as x + 4y

Since a pizza costs $12.75, hence x = $12.75

The function becomes 12.75 + 5y

f(y) = 12.75 + 5y

To get the total cost if one drink costs $3, hence;

f(3) = 12.75 + 5(3)

f(3) = 12.75 + 15

f(3) = 27.75

Hene the total cost, if one drink costs $3 is $27.75

Learn more on function here: brainly.com/question/9418047

7 0
2 years ago
On the first day of travel, a driver was going at a speed of 40 mph. The next day, he increased the speed to 60 mph. If he drove
gogolik [260]

Velocity, distance and time:

This question is solved using the following formula:

v = \frac{d}{t}

In which v is the velocity, d is the distance, and t is the time.

On the first day of travel, a driver was going at a speed of 40 mph.

Time t_1, distance of d_1, v = 40. So

v = \frac{d}{t}

40 = \frac{d_1}{t_1}

The next day, he increased the speed to 60 mph. If he drove 2 more hours on the first day and traveled 20 more miles

On the second day, the velocity is v = 60.

On the first day, he drove 2 more hours, which means that for the second day, the time is: t_1 - 2

On the first day, he traveled 20 more miles, which means that for the second day, the distance is: d_1 - 20

Thus

v = \frac{d}{t}

60 = \frac{d_1 - 20}{t_1 - 2}

System of equations:

Now, from the two equations, a system of equations can be built. So

40 = \frac{d_1}{t_1}

60 = \frac{d_1 - 20}{t_1 - 2}

Find the total distance traveled in the two days:

We solve the system of equation for d_1, which gets the distance on the first day. The distance on the second day is d_2 = d_1 - 20, and the total distance is:

T = d_1 + d_2 = d_1 + d_1 - 20 = 2d_1 - 20

From the first equation:

d_1 = 40t_1

t_1 = \frac{d_1}{40}

Replacing in the second equation:

60 = \frac{d_1 - 20}{t_1 - 2}

d_1 - 20 = 60t_1 - 120

d_1 - 20 = 60\frac{d_1}{40} - 120

d_1 = \frac{3d_1}{2} - 100

d_1 - \frac{3d_1}{2} = -100

-\frac{d_1}{2} = -100

\frac{d_1}{2} = 100

d_1 = 200

Thus, the total distance is:

T = 2d_1 - 20 = 2(200) - 20 = 400 - 20 = 380

The total distance traveled in two days was of 380 miles.

For the relation between velocity, distance and time, you can take a look here: brainly.com/question/14307500

3 0
2 years ago
Three times a number, x, increased by four is equal to five times the number, x. Which equation can be used to solve
ikadub [295]

Answer:

=5x⁴+3x

=-2x+4

=8x+4

=-2x-4

Step-by-step explanation:

hope it's help

7 0
2 years ago
Other questions:
  • HELPPPP! PLEASE EXPLAIN!!!
    5·1 answer
  • Two train whistles have identical frequencies of 180 Hz. When one train is at rest in the station, sounding its whistle, a beat
    5·1 answer
  • PLEASE HELP ME WITH THIS PROBLEMS 2.
    11·1 answer
  • Just one, help me it’s just one easy SIMPLE prob
    8·1 answer
  • What percent of 120 is 78?
    7·2 answers
  • Asap!! help me giving brainliest! No guessing.
    6·2 answers
  • The sum of the two shorter sides of a triangle must be equal to the longest side to be classified as a triangle
    11·1 answer
  • The function f(x) varies directly with x and f(x) = 12 when x = ¼. What is f(x) when x = 1/8?
    8·1 answer
  • A rectangle has a perimeter of 14 cm and length x cm. show that the width of the rectangle is (7-x)cm and hence that the area A
    7·2 answers
  • Help me to get Brainliest
    5·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!