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Free_Kalibri [48]
3 years ago
5

How many five-card hands (drawn from a standard deck) contain exactly three fives? (A five-card hand is any set of five differen

t cards.)
Mathematics
1 answer:
VashaNatasha [74]3 years ago
4 0

Answer:

4512.

Step-by-step explanation:

We are asked to find the number of five-card hands (drawn from a standard deck) that contain exactly three fives.

The number of ways, in which 3 fives can be picked out of 4 available fives would be 4C3. The number of ways in which 2 non-five cards can be picked out of the 48 available non-five cards would be 48C2.

4C3=\frac{4!}{3!(4-3)!}=\frac{4*3!}{3!*1}=4

48C2=\frac{48!}{2!(48-2)!}=\frac{48*47*46!}{2*1*46!}=24*47=1128

We can choose exactly three fives from five-card hands in 4C3*48C3 ways.

4C3*48C3=4*1128=4512

Therefore, 4512 five card hands contain exactly three fives.

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Let's take a look at the first few numbers in the sequence based on the given rule:

a_{1}=6\\ a_{2}= \frac{1}{4}(6)\\ a_3= \frac{1}{4}\big( \frac{1}{4}\big)6= \big(\frac{1}{4}\big)^2(6)\\ a_{4}= \frac{1}{4} \big(\frac{1}{4}\big)^26= \big(\frac{1}{4}\big)^3(6)

Inspecting this pattern it seems like the power \frac{1}{4} is being raised to is always one less than the number of the sequence, so if we were on the nth number in the sequence, that part of the expression would be \big(\frac{1}{4} \big)^{n-1}. We also know that we'll be multiplying whatever we get from that by 6, so we can write the full explicit rule for our sequence as

a_n=6\big( \frac{1}{4} \big)^{n-1}

Where a_n is the nth number in our sequence.
8 0
3 years ago
What is the LCM of 28,40, and 144
Simora [160]

Answer:

2 x 2 x 7 = 28 2 x 2 x 2 x 5 = 40 2 x 2 x 2 x 2 x 3 x 3 = 144 2 x 2 x 2 x 2 x 3 x 3 x 5 x 7 = 504... ... The LCM of the given two numbers using prime factorization is 36 ...

Step-by-step explanation:

8 0
3 years ago
Hello people ~
Luden [163]

Cone details:

  • height: h cm
  • radius: r cm

Sphere details:

  • radius: 10 cm

================

From the endpoints (EO, UO) of the circle to the center of the circle (O), the radius is will be always the same.

<u>Using Pythagoras Theorem</u>

(a)

TO² + TU² = OU²

(h-10)² + r² = 10²                                   [insert values]

r² = 10² - (h-10)²                                     [change sides]

r² = 100 - (h² -20h + 100)                       [expand]

r² = 100 - h² + 20h -100                        [simplify]

r² = 20h - h²                                          [shown]

r = √20h - h²                                       ["r" in terms of "h"]

(b)

volume of cone = 1/3 * π * r² * h

===========================

\longrightarrow \sf V = \dfrac{1}{3}  * \pi  * (\sqrt{20h - h^2})^2  \  ( h)

\longrightarrow \sf V = \dfrac{1}{3}  * \pi  * (20h - h^2)  (h)

\longrightarrow \sf V = \dfrac{1}{3}  * \pi  * (20 - h) (h) ( h)

\longrightarrow \sf V = \dfrac{1}{3} \pi h^2(20-h)

To find maximum/minimum, we have to find first derivative.

(c)

<u>First derivative</u>

\Longrightarrow \sf V' =\dfrac{d}{dx} ( \dfrac{1}{3} \pi h^2(20-h) )

<u>apply chain rule</u>

\sf \Longrightarrow V'=\dfrac{\pi \left(40h-3h^2\right)}{3}

<u>Equate the first derivative to zero, that is V'(x) = 0</u>

\Longrightarrow \sf \dfrac{\pi \left(40h-3h^2\right)}{3}=0

\Longrightarrow \sf 40h-3h^2=0

\Longrightarrow \sf h(40-3h)=0

\Longrightarrow \sf h=0, \ 40-3h=0

\Longrightarrow \sf  h=0,\:h=\dfrac{40}{3}<u />

<u>maximum volume:</u>                <u>when h = 40/3</u>

\sf \Longrightarrow max=  \dfrac{1}{3} \pi (\dfrac{40}{3} )^2(20-\dfrac{40}{3} )

\sf \Longrightarrow maximum= 1241.123 \ cm^3

<u>minimum volume:</u>                 <u>when h = 0</u>

\sf \Longrightarrow min=  \dfrac{1}{3} \pi (0)^2(20-0)

\sf \Longrightarrow minimum=0 \ cm^3

6 0
2 years ago
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Which is the sum of 3/10 and 1/3
Ilya [14]

Step-by-step explanation:

\:  \:  \:  \:  \: \frac{3}{10}  +  \frac{1}{3}  \\  \\  =  \frac{3 \times 3}{10 \times 3}  +  \frac{1 \times 10}{3 \times 10}  \\  \\  =  \frac{9}{30}  +  \frac{10}{30}  \\ \\   =  \frac{19}{30}  \\  \\ so \: as \: you \: see \: here \: we \: should \: put \: same \: denominator \\ \: for \: both \: fractions. \: hope \: this \: helps.. \\ good \: luck

8 0
3 years ago
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Reptile [31]

The variance of a distribution is the square of the standard deviation

The variance of the data is 2.2

<h3>How to calculate the variance</h3>

Start by calculating the expected value using:

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So, we have:

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This gives

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Next, calculate E(x^2) using:

E(x^2) = \sum x^2* P(x)

So, we have:

E(x^2) = 1^2 * 0.3 + 2^2* 0.2 +3^2 * 0.2 + 4^2 * 0.1 + 5^2 * 0.2

E(x^2) = 9.5

The variance is then calculated as:

Var(x) = E(x^2) - (E(x))^2

So, we have:

Var(x) = 9.5 - 2.7^2

Var(x) = 2.21

Approximate

Var(x) = 2.2

Hence, the variance of the data is 2.2

Read more about variance at:

brainly.com/question/15858152

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2 years ago
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