We wish to choose 7 cards from a usual deck of 52 playing cards. In how many ways can this be done if we are required to choose the cards in the following ways?
(a) with no restriction.
(b) all cards come from the same suit.
(c) exactly 3 Aces and exactly 3 Kings are chosen.
(d) all 7 cards have values between 2 and 7 inclusive.
(e) all 7 cards all have different values (where Jacks are different from Queens, etc.).

Answers

Answer 1

In a usual deck of 52 cards, there are 4 suits with 13 card for each suit.

(a) with no restriction.
If there is no restriction and different order is not important, then the possible way should be: 52!/ (52-7)!7!= 52!/45!7!= (52* 51* *50*49*48*47*46)/(7*6*5*4*3*2*1) = 674274182400/5040= 133,784,560 ways

(b) all cards come from the same suit.
If all card has to come from the same suit, that means you will only have 13 possible cards for each suit.
Then the possible ways for each suit: 13!/(13-7)!7!= (13*12*11*10*9*8*7)/ (7*6*5*4*3*2)= 8648640/5040= 1716 ways per suit.
Since there are 4 suits then the possible way= 1716 * 4= 6864

(c) exactly 3 Aces and exactly 3 Kings are chosen.
In this case, we will have 3 aces, 3 kings, and one random card. There are 4 aces/kings in one deck and we have to choose 4.
The possible ways for 3 aces or 3 kings would be: 4!/3!(4-3)!= 4 ways.
After choosing the 3 aces and 3 kings, the deck should have 52-3-3 = 46 card left. That mean there will be 46 possible ways for the random card.
The total possibilities should be: 4 * 4* 46= 736 ways

(d) all 7 cards have values between 2 and 7 inclusive
There are four cards with each value. The value between 2 and 7 inclusive should be 7-2+1= 6 different value. Since each value has 4 cards and we have 6 different value, then the total possible card is 6*4= 24
Then, we take 7 cards from those 24 cards. The possible should be: 24!/7!(24-7)!= (24*23*22*21*20*19*18) / 5040= 1744364160/5040= 346104

(e) all 7 cards all have different values (where Jacks are different from Queens, etc.).
There is 4 card with same value. For the first take, you will have 52 different ways. But for the second take, you will only have 52-4=48 different ways. That was because you can use:
1. one card that you take earlier.
2. three cards with the same value.
Then the possible ways become like this: 52*48*44*40*36*32*28/ 7!= 28,114,944

Related Questions

In a fraternity with 37 members, 18 take mathematics, 5 take both mathematics and art history, and 8 take neither mathematics nor art history. How many take art history but not mathematics?

Answers

fraternity of 32 members with 8 taking neither math or physics
Of the 24, 5 took both. 24 - 5 = 19 taking one. 18 taking just Math.
Just 1 took only Physics.

checking our answer:
just Math 18
just Physics 1
both 5
neither 8
total 32
just physics = 32 - 8 -5 -18 = 1

Answer:

Fraternity of 32 members with 8 taking neither math or physics

Of the 24, 5 took both. 24 - 5 = 19 taking one. 18 taking just Math.

Just 1 took only Physics.

Step-by-step explanation:

(1 point) consider the function f(t)=⎧⎩⎨⎪⎪⎪⎪0,−5,−6,6,t<00≤t<11≤t<7t≥7;f(t)={0,t<0−5,0≤t<1−6,1≤t<76,t≥7; 1. write the function in terms of unit step function

Answers

Final answer:

The function f(t) can be represented in terms of the unit step function as f(t) = -5u(t) - u(t-1) + 6u(t-7), with each term turning on at specific breakpoints.

Explanation:

To write the function f(t) in terms of the unit step function, you can think of f(t) as a piecewise function that changes values at various breakpoints. The unit step function, denoted as u(t), is defined to be 0 when t < 0 and 1 when t ≥ 0. For the given function f(t), we can write it using the unit step function as follows:

f(t) = -5u(t) -u(t-1) + 6u(t-7)

This representation uses the properties of the unit step function to 'turn on' different function values at specified points. Each term corresponds to a segment of the piecewise function, with the coefficients representing the value of f(t) within each segment.

sales price is 67.50 sales tax is 5% how much is the total amount paid

Answers

$70.88

x/67.5=5/100
cross multiply
100x=337.5
divide 100
x=3.375 or 3.38

67.50+3.38= 70.88

Paul is riding his bike 48 miles to the beach. His average rate of speed is 13.8 miles per hour. Paul stopped for a snack after 3 hours. How many miles does he have to go after his snack? Round your answer to the nearest tenth if necessary.

Answers

the answer is 6.6 miles

13.8 miles per hour x 3 hours = 41.4 miles ( what he already rode)

48 - 41.4 = 6.6 more miles to go

Find equations for the horizontal and vertical tangent lines to the curves

Answers

That's where slope is 0, any line tangent at that point will be horizontal: when x=3 or when x=−1. Then find the corresponding y value in the first equation: f(x)=(x−2)(x2−x−11), to determine the two points: (x1,y1),(x2,y2) where the line tangent to f(x) is horizontal

:what does this problem solve as
1/2(4x−5)+52=10

Answers

I'm pretty sure its x=−79/4

The difference in elevation of a helicopter and a submarine is 18 1/2 meters. The elevation of the submarine is -7 3/4 meters. Write and solve an equation to find the elevation h (in meters) of the helicopter

Answers

The helicopter is above sea level and the sub below sea level.

One way in which to write an equation here is |-7 3/4 meters| + h = 18 1/2.

let the elevation of the submarine = s, and elevation of the helicopter = h

the difference in elevation = h - s = 18 1/2

the question gives us the value of s: -5 3/4

so sub that into our first equation

h - (-5 3/4) = 18 1/2
h + 5 3/4 = 18 1/2
h = 18 1/2 - 5 3/4
h = 12 3/4

hope it helps :)

A certain recipe calls for 2 3/4 cups of sugar.if the recipe is to be doubled, how much sugar should be used

Answers

2 and 3/4 + 2 and 3/4 is 5 and 1/2.

the temperature of a pot of water is shown. the temperature of the water changed -2.5 degree F per minute. What was the temperature after 20 minutes?

Answers

-50 degree f because speed is -2.5 times time (20)
=-50

At a pet store, Davina counted 12 parrots out of 20 birds. What is an equivalent ratio of parrots to birds in the pet store?

Answers

An equivalent ratio of parrots to total birds would be 3:5. Hope this helps!

The ratio that you are looking for corresponds to 12:20 is 3:5.

You are debating about whether to buy a new computer for $800.00 or a refurbished computer with the same equipment for $640.00. If you are paid $10.00/hour and your deductions are FICA (7.65%), federal tax withholding (12%), and state tax withholding (8%), how many hours do you have to work to pay for the new computer?

A) 111 hours

B) 89 hours

C) 98 hours

D) 87 hours

Answers

First calculate all the deduction

FICA = $10 per hour x 0.0765 = $0.77 per hr

Fed tax = $10 per hr x 0.12 = $1.2 per hr

State tax = $10 per hr x 0.08 = $0.8 per hr

Remaining pay = 10 – 0.77 – 1.2 – 0.8 = $7.23 per hour

Time required = $640 / $7.23 per hr

Time required = 89 hrs

After calculating deductions from the gross wage of $10.00, the net income per hour is $7.235. To buy the new computer for $800, we divide the cost by the net wage, resulting in approximately 110.58 hours. Therefore, the student would need to work about 111 hours to afford it.

To calculate the number of hours needed to work to pay for the new computer, we must first determine the net income per hour after all the deductions. With a gross wage of $10.00 per hour, the deductions would be:

FICA: 7.65% of $10.00 = $0.765Federal Tax: 12% of $10.00 = $1.20State Tax: 8% of $10.00 = $0.80

This totals $2.765 in deductions per hour. Subtracting this from the gross wage gives a net income per hour:

$10.00 - $2.765 = $7.235.

To afford the new computer costing $800, we divide the cost of the computer by the net income per hour to find the total number of hours needed:

$800.00 / $7.235 ≈ 110.58 hours.

Since we cannot work a fraction of an hour, we will need to round up to the nearest whole number, which suggests that the student would need to work approximately 111 hours to afford the new computer.

how many factors are in the expression x(1+y)^z
a. 3
b.2
c.0
d.1

Answers

There are two factors: ( x ) and (1+y). So option (b) is correct.

To find the factors in the expression ( x(1+y)^z ), we break it down into its components.

1. ( x ) - This is a factor because it's multiplied by the rest of the expression.

2. (1+y) - This is also a factor because it's part of the expression being raised to the power of $ z $.

3. ( z ) - The exponent does not affect the number of factors, as it's simply the power to which the expression is raised.

So, there are two factors: ( x ) and (1+y).

Detailed calculation:

( x ) is multiplied by the entire expression, so it's a factor.

(1+y) is a factor because it's part of the expression.

( z ) is an exponent and doesn't count as a separate factor.

Thus, the answer is (b) 2.

What is the quotient of d-2 / d4-6d3+d+17=
A. d3− 4d2 + 9d + 35 B. d3− 8d2 + 17d –17 C. d3− 8d2 + 16d − 31 R 79 D. d3– 4d2– 8d –15 R–13

Answers

Refer to the diagram below

We have (d⁴-6d³+d+17) ÷ (d-2)

Step 1:
Multiply d by a term that would give result of d⁴
The aim of this working out is to eliminate on one term at a time, starting from the highest polynomial

We have d⁴ ÷ d = d³ (Write this on the top)

Step 2:
Multiply back d³ by (d - 2) to get d⁴ - 2d³

Step 3:
Then subtract (d⁴ - 2d³) from (d⁴ - 6d³ + 0d² + d + 17)
Notice that the term d⁴ is now eliminated
We have (-6d³ - -2d³) = (-4d³)
Then pull down the third term which is 0d²
We have the sum -4d³ + 0d²

Step 4:
Divide (-4d³) by (d) to get (-4d²)
Write (-4d²) next to (d³) on the top

Step 5:
Then multiply back (-4d²) by (d - 2) to get (-4d³+8d²)

Step 6:
Subtract (-4d³+8d²) from (-4d³ + 0d²) to get (-8d²) then pull down the next term (d) to make the sum (-8d² + d)
Notice that the term -4d³ is now eliminated

Step 7:
Divide -8d² by d to get -8d
Write -8d on the top, next to d³ - 4d²

Step 8:
Multiply back (-8d) by (d - 2) to get (-8d² + 16d)

Step 9:
The subtract (-8d² + 16d) from (-8d + d) to get (-15d + 17)
Notice that the term (-8d²) has been eliminated
The last step is to divide -15d by d to get -15
Write -15 on the top next to the sum d³ - 4d² - 8d
Then multiply -15 by (d-2) to get -15d + 30
Subtract (-15d + 30) from (-15d + 17) to get a REMAINDER of 13

The answer is d³ - 4d² - 8d - 15 R 13
OPTION D

Answer:

Step-by-step explanation:

hey can the other person get brainliest. thanks

Which group of 3 cards would allow Carlos to win with 21 points?

A. 6 of clubs, 7 of spades, 8 of diamonds
B. Ace of hearts, Ace of diamonds, 10 of hearts
C. Jack of clubs, 7 of clubs, 4 of spades
D. 8 of clubs, 9 of hearts, 9 of spades

Answers

A and C equal 21
☺☺☺☺

mrs. baker purchased a number of juice packs at a cost of $0.30 each and a loaf of bread that cost $1.19. the total cost of her purchases was 2.99. which equation would you use the determine how many juice packs mrs. baker purchased

Answers

0.30x + 1.19 = 2.99

hope this helps

Answer:

0.30j + 1.19 = 2.99

Step-by-step explanation: Hope this helps! :D

A condo in Culver City, California, listed for $1.4 million with 20% down and financing at 5% for 30 years. What's the monthly payment?

Answers

$6,387.40 per month. The following formula is used to calculate the fixed monthly payment (P) required to fully amortize a loan of L dollars over a term of n months at a monthly interest rate of c. P = L[c(1 + c)^n]/[(1 + c)^n - 1]

Answer:

Step-by-step explanation:

$6,014.40

A laptop has a listed price of 715.95 before tax. If the sale tax rat is 9.25%, find the cost of the laptop with the sales tax included. Round your answer to the nearest cent, as necessary.

Answers

$715.95 + 9.25% (66.22)= $782.18

(40 points + Brainliest for right answer).

Answers

the correct answer is C

Atif has a z score of 1.02 from a converted iq score. from a z table, the percentage between his score and the mean is 34.61%. what percent of people have iqs at or lower than atif's score?

Answers

Answer: 84.61%

Explanation.
Refer to the diagram shown below.

When the z-score is z = 0, then the area under the distribution curve is 50%, which corresponds to the mean, μ.

The given z-score is greater than zero, so the shaded area is 34.61%. to the right of z=0
The area to the left of z = 1.02 is 50% + 34.61% = 84.61%.
This percentage represents the people with IQs at or lower than Atfif's score.

What is the least possible sum of 24

Answers

What do you mean? By the least possible sum of 24

Prove the identity csc^2xsec^2x=sec^2x+csc^2x

Answers

[tex]\bf csc(\theta)=\cfrac{1}{sin(\theta)} \qquad % secant sec(\theta)=\cfrac{1}{cos(\theta)} \\\\\\ and\qquad sin^2(\theta)+cos^2(\theta)=1 \\\\ -------------------------------\\\\ csc^2(x)sec^2(x)=sec^2(x)+csc^2(x)\\\\ -------------------------------\\\\ sec^2(x)+csc^2(x)\implies \cfrac{1}{cos^2(x)}+\cfrac{1}{sin^2(x)}\implies \cfrac{sin^2(x)+cos^2(x)}{cos^2(x)sin^2(x)} \\\\\\ \cfrac{1}{cos^2(x)sin^2(x)}\implies \cfrac{1}{cos^2(x)}\cdot \cfrac{1}{sin^2(x)}\implies sec^2(x)csc^2(x)[/tex]

The shortest side of a right triangle measures 11m. The lengths of the other two sides are consecutive integers. Find the lengths of the other two sides.

Answers

Interesting question. Let the 2 unknown sides be x and x+1.
Then (11 m)^2 + x^2 = sum of the squares of the 2 shortest sides
= (x+1)^2

121 + x^2 = x^2 + 2x + 1. Then 121 = 2x + 1, and 2x = 120, or x = 60.

Then the hyp. has length 60+1= 61.

We must check these results. Using the Pyth. Thm. (a^2 + b^2 = c^2),

11^2 + 60^2 = 61^2
121 + 3600 = 3721 This is true, so our answers are correct.

The longer side is 60 and the hyp. has length 61.

Silver is a white metal and is an excellent conductor of heat and electricity. The density of silver is 10.49 g/cm3. Silver does not react with water but it does react with nitric acid. Silver tarnishes when exposed to air or light. What is a physical property of silver?

Answers

A physical property is that it is a white metal. The other descriptions are all chemcial properties such as how it reacts and conducts. Density could also be another answer as it is also physical.

density because everything else would fall under the category of chemical properties instead of physical properties and the other 3 properties would result in a chemical change while density would not

Over a 12 hour period from 8 pm to 8 am the tempeture fell a steady rate 8°F to 16°F if the temperature fell at the same rate every hour what was the temperature at 4 am

Answers

What we know:
12 hour period from 8pm to 8am
temperature dropped from 8°F to 16°F from 8pm to 8am

We need to find temperature at 4 am.

We can start by setting up points:
8pm is are starting point with 8°F, we can express it as (0,8), 0 represents initial time from 0 to 12 hour span.
8am is the ending point with 16°F, we can express it as (12,16), 12 represents the end time of 0 to 12 hours span.
We will use these points to find slope.
slope=m=(16-8)/(12-0)=8/12=2/3

Now, we can set up an expression to find any temperature at a specific time. Aslo, x represents the hours not the the specific time of 4am. We will use 8 since 4am is the 8th hour of the 12 hour span. Using slope of 2/3 and the y intercept of (0,8) since we were already at 8°F at the initial time of 0 we have the function:

f(x)=2/3x+8

f(8)=2/3(8)+8= 40/3≈13.3°

Lucy is thinking of a number.The number is greater than two hundred twenty- five. her number is less than 2 hundred 2 tens 7 ones. what is lucy's number?

Answers

226 is Lucy’s number

226
I am trying to add more characters

About 19/20 of celery is water. What percent is this

Answers

Percents have to fit into 100.

So, 20x5= 100
19x5=95

So, your answer is: 95%

Hello There!

19/20
You need to get the denominator to 100.
20 x 5 = 100
19 x 5 = 95

It is 95%

Hope This Helps You!
Good Luck :)

- Hannah ❤

Please help me on 3 and 4 please thanks ❤️

Answers

For the perimeter, we can literally count 8 sides, and since we have 6 inches per side, 6 added to itself 8 times or 6*8=48

Since we know that there are 12 inches in a foot, and 12 feet, we have
[tex] \frac{12inches}{1foot} * 12feet=144 inches[/tex]

Feel free to ask with any further questions!

Which equation represents the linear function that is shown on the graph below?

ANSWERS:

y= 1/2x +4

y= 1/2x -2

y= 4x -2

y= -2x +4

Answers

Answer is choice B) y = 1/2x - 2

which is the same as y = (1/2)x - 2 or can be written as [tex]y = \frac{1}{2}x - 2[/tex]

If you wanted the decimal form version, then the equation would be y = 0.5x - 2
------------------------------------------
------------------------------------------

Explanation:

The four points on this line are
(4,0)
(6,1)
(8,2)
(10,3)

Let's pick two of these points to find the slope.

The two points I'll pick are (4,0) and (6,1)

The first point is (4,0) so (x1,y1) = (4,0) meaning that
x1 = 4
y1 = 0

The second point is (6,1) so (x2,y2) = (6,1) meaning that
x2 = 6
y2 = 1

Using the slope formula, we find...
m = (y2 - y1)/(x2 - x1)
m = (1-0)/(6-4)
m = 1/2
m = 0.5

The slope in fraction form is 1/2. This means that each time we move up 1 unit, we move to the right 2 units.
slope = rise/run = 1/2
Rise = 1
Run = 2

The fraction 1/2 is equivalent to the decimal form 0.5

------------------------------------------

Let's use the slope m = 0.5, along with the point (x,y) = (4,0) to find the y intercept b

y = m*x + b
y = 0.5*x + b
0 = 0.5*4 + b
0 = 2 + b
0 - 2 = 2 + b - 2
-2 = b
b = -2
The y intercept is -2 meaning that the line crosses the vertical y axis at -2.

Therefore the equation is y = 0.5x - 2 after we update y = mx+b

This is the same as y = (1/2)x - 2 or y = 1/2x - 2 which is why choice B is the answer. The parenthesis help make it clear that x is not in the denominator; though technically they aren't needed.

The equation y = (1/2)x - 2.

What is a function?

A relation between a collection of inputs and outputs is known as a function. A function is a connection between inputs in which each input is connected to precisely one output. Each function has a range, codomain, and domain.

We have points from the graph (6,1) and (8,2).

The slope of the equation is;

m = (2-1) / (8-6)

m = 1/2

And the y-intercept;

We have the slope-intercept form of the equation,

y = mx + b

y = x/2 + b

To find the y-intercept, we substitute the point (6,1)

b = -2

Therefore, we have the equation of the linear function y = (1/2)x - 2.

To learn more about the function;

brainly.com/question/28303908

#SPJ6

50 POINTS!!!

Suzanne ran 3 3/4 miles in the morning and 4 1/2 miles in the afternoon. How many kilometers did she run altogether? Assume 1 mile = 1.6 kilometers.

Answers

4 1/2 = 18/4
3 3/4 = 15/4

15/4 + 18/4 = 8 1/4 or 33/4

33/4 * 1.6/1 = 52.8/4 = 13.2

When assuming 1 mile equals 1.6 kilometers, Suzanne ran 13.2 kilometers in all.

Answer:- Total distance traveled by Suzanne in kilometers=13.2 kilometers.

Explanation:-

Distance traveled by Suzanne in the morning=[tex]3\frac{3}{4}\text{ miles}[/tex]

Distance traveled by Suzanne in the evening=[tex]4\frac{1}{2}\text{ miles}[/tex]

Total distance traveled by Suzanne=[tex][3\frac{3}{4}+4\frac{1}{2}\text{ miles}[/tex]

[tex]=\frac{15}{4}+\frac{9}{2}\text{ miles}[/tex]

[tex]=\frac{15+18}{4}\text{ miles}[/tex]

[tex]=\frac{33}{4}\text{ miles}[/tex]

[tex]=8.25\text{ miles}[/tex]

Total distance traveled by Suzanne in kilometers=[tex]8.25\times1.6\text{ kilometers}=13.2\text{ kilometers}[/tex]

Total distance traveled by Suzanne in kilometers[tex]=13.2\text{ kilometers}[/tex]

Find f. f '''(x) = cos(x), f(0) = 4, f '(0) = 2, f ''(0) = 6

Answers

Given that every integration has constant C, and is solved by the 0 conditions stated in the problem;
f'''(x) = Cos(x)
f''(x) = sin(x) + 6 after f''(0) = 6
f'(x) = -cos(x) + 6x + 2 after f'(0) = 2
f(x) = -sin(x) + 3x^2 + 2x + 4 after f(0) = 4

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