¹³C₈ + ¹³C₉ as a single term from Pascal's Triangle is ¹⁴C₉
What is Pascal's triangle?
Pascal's triangle is a triangle written in such a way that it forms the coefficients of a binomial expansion. The coefficients of the terms are gotten through combination.
What is combination?Combination is the number of ways r in which n objects can be selected. It is given by ⁿCₓ = n!/x!(n - x)!
How to write a single term from Pascal's Triangle in the form t_nr = t_13,8 + t_13,9.Since we have ¹³C₈ + ¹³C₉ and we want to write it as a single term, we have that
¹³C₈ = 13!/8!(13 - 8)! = 13!/8!5! and ¹³C₉ = 13!/9!(13 - 9)! = 13!/9!4!So, ¹³C₈ + ¹³C₉ = 13!/8!5! + 13!/9!4!
= 13!/(8! × 5 × 4!) + 13!/(9 × 8! × 4!)
= 13!/8!4![1/5 + 1/9]
= 13!/8!4! × [(9 + 5)/45]
= 13!/8!4! × 14/45]
= 13!/8!4! × 14/(9 × 5)]
= 14 × 13!/8! × 9 × 4! × 5)]
= 14!/9!5!
= 14!/9!(14 - 9)!
= ¹⁴C₉
So, ¹³C₈ + ¹³C₉ as a single term from Pascal's Triangle is ¹⁴C₉
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A container contains 20 liters of milk and another container contains 20 liters of syrup. One liter was taken from each vessel and poured into another vessel. After doing this twice, determine the ratio of milk and syrup in the two types of mixture.
Answer:
Step-by-step explanation:
20 liters -1 liter = 19 liters of milk
20 liters -1 liter = 19 liters of syrup
This was done twice so therefore;
19 liters -1 liter = 18 liters of milk
19 liters -1 liter = 18 liters of syrup
Therefore the ratio comes to 18:18
18 goes into 18 1 and the same for the other 18. Therefore the ratio comes to ;
1:1
Your final answer is 1:1
Can someone help me out please
By just adding the given areas, we conclude that the definite integral is equal to 3.026
How to calculate the definite integral?
The integral will be equal to the sum of the four areas between points a and c.
The areas above the horizontal axis are positive areas, these are A and C.
While the areas below the horizontal axis are negative areas, these are B and D.
Then the definite integral will be:
I = A - B + C - D
Here we know that:
A = 1.311
B = 2.229
C = 5.545
D = 1.601
Replacing that, we conclude that the definite integral is
I = 1.311 - 2.229 +5.545 - 1.601 = 3.026
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3. The function defined by m(h) = 300 * (3/4) ^ h represents the amount of a medicine, in milligrams, in a patient's bodyrepresents the number of hours after the medicine is administered . a. What does m(0.5) represent in this situation?
This means that after 0.5 hours, 259.81 milligrams of the medicine remain on the patient's body.
What does m(0.5) represent in this situation?We know that the exponential function defined by:
[tex]m(h) = 300*(3/4)^h[/tex]
Represents the amount of a mediciene, in milligrams, after a number of hours h.
We want to see what does m(0.5) represent. This is the exponential function evaluated in h = 0.5
Then, it just represents the amount of medicine, in miligrams, in a patient's body present 0.5 hours after the medicine was administered.
Replacing h = 0.5 we get:
[tex]m(0.5) = 300*(3/4)^{0.5} = 259.81[/tex]
This means that after 0.5 hours, 259.81 milligrams of the medicine remain on the patient's body.
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Harvey is analyzing the production cost of a new product launched by his company. The initial production cost was $1,050. The production cost is at its lowest amount, $250, for 200 items, and thereafter increases as the number of items increases. Which of the following graphs represents the production cost of the product?
It would be the bottom left graph
Initial production: 1050
so the graph will be started at 1050 which makes top right graph incorrect
Next, the money at the lowest was 250 which makes the top left and bottom right incorrect.
leaving only the bottom right graph to be correct
Answer:
Graph Y
Step-by-step explanation:
Given information:
Initial production cost = $1,050Lowest production cost = $250 for 200 itemsProduction cost increases after 200 items.The x-axis shows number of items in hundreds.The y-axis shows the cost in dollars.The initial production cost is when the number of items is zero.
Therefore, the y-intercept of the graph will be (0, 1050).
The lowest production cost is the minimum point of the curve.
Therefore, the vertex of the graph will be (2, 250).
The only graph that satisfies these conditions is graph Y (attached).
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Suppose you are building a storage box of volume 4368in^3. the length of the box will be 24 in. the height of the box will be 1 in. more than its width. find the height and the width of the box.
Answer:
height: 14 incheswidth: 13 inchesStep-by-step explanation:
The volume formula can be used to find the height and width of a box with volume 4368 in³ and height 1 in greater than width.
SetupThe volume formula is ...
V = LWH
Substituting given information, using w for the width, we have ...
4368 = (24)(w)(w+1)
SolutionWe want to find the value of w.
182 = w² +w . . . . . . . . divide by 24
182.25 = w² +w +0.25 = (w +0.5)² . . . . . . add 0.25 to complete the square
13.5 = w +0.5 . . . . . . . . take the positive square root
w = 13 . . . . . . . . . . . . subtract 0.5
h = w+1 = 14
The height of the box is 14 inches; the width is 13 inches.
__
Additional comment
By "completing the square", we can arrive at the exact dimensions of the box, as we did above. Note that we only added 0.25 to the equation to do this.
For numbers close together, the geometric mean (root of their product) is about the same as the arithmetic mean (half the sum):
[tex]\sqrt{w(w+1)}\approx\dfrac{w+(w+1)}{2}=w+\dfrac{1}{2}\\\\w\approx\sqrt{182}-\dfrac{1}{2}\approx12.99[/tex]
Using this approximation to arrive at the conclusion w=13 saves the steps of figuring the value necessary to complete the square, then adding that before taking the root.
what do I check off
Mrs. Riley recorded this information from a recent test taken by all of her students. Using the data, what was the average percent score for these $100$ students?
PLEASE HELP
Using the given data, the average percent score for the 100 students is 77[tex]\%[/tex].
Average, also known as mean, is one of the measures of central tendency and is the ratio of the sum of all observations to the total number of observations. It is very useful to summarize a probability distribution.
Let the average percent score for the 100 students be N.
As it is known that the average is the ratio of the sum of all observations to the total number of observations, therefore:
[tex]N = \dfrac{100\times7+90\times18+80\times35+70\times25+60\times10+50\times3+40\times2}{100}[/tex]
[tex]= \dfrac{7700}{100}\\= 77[/tex]
Thus, the average percent score for the 100 students, using the given data is calculated as 77[tex]\%[/tex].
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The complete question is as follows:
Mrs. Riley recorded this information from a recent test taken by all of her students. Using the data, what was the average percent score for these 100 students?
[tex]\%[/tex] Score Number of students
100 7
90 18
80 35
70 25
60 10
50 3
40 2
Find the area of the polygon with vertices
A(0,5), B(0,2), C(-3,2), and D(-3,5)?
Answer: 9
Step-by-step explanation:
assuming that each box is 1 square inch then, you would get 9 square inches.
0,5 go down three to get 0,2
0,2 go left three to get -3,2
-3,2 go up three to get -3,5
-3,5 go right three to get 0,5
so you end up with a 3-by-3 box
A choir has 3 spots open for altos, and 8 altos are interested in them. in how many ways can the open spots be filled? 24 56 112 336
Answer:
56
Step-by-step explanation:
combinations
C(8,3)
The sum of five consecutive positive integers is 2020. Find the largets of these numbers.
Answer:
406
Step-by-step explanation:
let the 5 consecutive positive integers be
n , n + 1 , n + 2 , n + 3 , n + 4 , then
n + n + 1 + n + 2 + n + 3 + n + 4 = 2020
5n + 10 = 2020 ( subtract 10 from both sides )
5n = 2010 ( divide both sides by 5 )
n = 402
then
largest integer = n + 4 = 402 + 4 = 406
Quick algebra 1 question for 10 points!
Only answer if you know the answer, quick shout-out to tariqareesha2 and MrBrainly, tysm for the help!
Answer:
C (Third option)
Step-by-step explanation:
The prime factorization of 675 is 5*5*3*3*3 or 5^2*3^3. With the square root, we take out all square integers, leaving (5*3)(sqrt3). The answer is 15(sqrt3) or C.
Break
√225×3√225√3√15²√315√3Option C
Devika buys 30 notebooks and 15 pens. Each book costs Rs40 and that of each pen is Rs15. Find out the total amount she spent.
Answer:
Rs 1425
Step-by-step explanation:
=(30×40) + (15×15)
=1425
PLS HELP ASAP??!! Pls help me with 7,8,8
Answer:
Step-by-step explanation:
1144
correct 0.34213 to three significant figures
Answer:
0.342
Step-by-step explanation:
round to the given amount of significant figures
If AB = 55, and BC = 49, find the length of the radius to the nearest tenth. Assume BC is tangent to Circle A.
Answer:
25.0
Step-by-step explanation:
Notice that points A, B, and C form a right triangle. This allows us to use the Pythagorean theorem, a^2 + b^2 = c^2, to find the missing side. AB is the hypotenuse, so substitute 55 for c, and BC is the leg, so substitute 49 into either a or b in the formula.
a^2 + b^2 = c^2
49^2 + b^2 = 55^2
b^2 = 3025 - 2401
b^2 = 624
b = sqrt (624)
b = 24.9799919936
pls help with math rn neeedddd
Answer:
40° to the left and 3/2 units down
Step-by-step explanation:
See attached image.
Help me in thisssssss
Answer:
34
Step-by-step explanation:
(a + [tex]\frac{1}{a}[/tex] ) = 6← square both sides
(a + [tex]\frac{1}{a}[/tex] )² = 6² ← expand left side using FOIL
a² + 1 + 1 + [tex]\frac{1}{a^2}[/tex] = 36
a² + 2 + [tex]\frac{1}{a^2}[/tex] = 36 ( subtract 2 from both sides )
a² + [tex]\frac{1}{a^2}[/tex] = 34
A boat took 4 hours to make a downstream
trip with a current of 6 km/h. The return trip
against the same current took 6 hours. Find
the speed of the boat in still water.
Answer:
30 km/h
Step-by-step explanation:
Let X km/h be the speed of the boat in still water
Let the distance covered each way be D km
Relative speed of boat downstream = X + 6
Relative speed of boat upstream = X - 6
Distance covered by boat when going downstream
D = 4(X + 6) = 4X + 24
Distance covered by boat upstream is the same as distance covered downstream
D = 6(X-6) = 6X -36
Setting the two X equations equal to each other gives us
6X - 36 = 4X + 24
Collecting like terms
6X - 4X = 24-(-36)
2X = 60
X = 30 km/hr which is the speed of boat in still water
Check
Downstream speed is 30+6 = 36 km/h and distance traveled downstream in 4 hours = 36 x 4 = 144km
Upstream speed is 30-6 = 24 km/h and it travels for 6 hours. Distance covered = 24 x 6 = 144 miles
So distance covered is same and hence check succeeds
hich is the best approximation for the solution of the system of equations?
y = A system of equations. y equals negative StartFraction 2 over 5 EndFraction x plus 1. y equals 3 x minus 2.x + 1
y = 3x – 2
the solution of the system of linear equations is:
x = 6/11
y = -4/11
How to solve the system of equations?
Here we have the system of equations:
y = (-5/2)*x + 1y = 3x - 2To solve the system, we can see that y is already isolated in both sides, then we get:
(-5/2)*x + 1 = 3x - 2
Now we can solve this for x:
(-5/2)*x + 1 = 3x - 2
2 + 1 = 3x + (5/2)*x
3 = (6/2)*x + (5/2)*x
3 = (11/2)*x
(2/11)*3 = x
6/11 = x
To get the value of y, we evaluate any of the lines in x = 6/11
y = 3*(6/11) - 2 = 18/11 - 2 = 18/11 - 22/11 = -4/11
Then the solution of the system of linear equations is:
x = 6/11
y = -4/11
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find the average rate of change of the function in the interval 6,13
Using it's concept, the average rate of change of the function on the interval [6,13] is given by:
[tex]r = \frac{f(13) - f(6)}{7}[/tex]
What is the average rate of change of a function?The average rate of change of a function is given by the change in the output divided by the change in the input. Hence, over an interval [a,b], the rate is given as follows:
[tex]r = \frac{f(b) - f(a)}{b - a}[/tex]
In this problem, we want to find the rate in the interval [6, 13], which is given as follows:
[tex]r = \frac{f(13) - f(6)}{13 - 6} = \frac{f(13) - f(6)}{7}[/tex]
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Quick algebra 1 question for 10 points!
Only answer if you know the answer, quick shout-out to tariqareesha2 and MrBrainly, tysm for the help!
Answer:
9 or 16... but I'm leaning more to 9
Step-by-step explanation:
Since the graph seems to be going up and down, it seems like its at its highest point and should go down now so it could be 9. But if not and its still going to go higher 16 is possible.
Ariel completed the work below to show that a triangle with side lengths of 9, 15, and 12 does not form a right triangle. 9 squared 15 squared = 12 squared. 81 225 = 144. 306 not-equals 144.
Answer:
Work is incorrect
Step-by-step explanation:
I'm assuming this question is asking whether the work is correct or not? In which case the work is not correct.
The Pythagorean Theorem states: [tex]a^2+b^2=c^2[/tex] where c=hypotenuse, and "a" and "b" are the other two sides. The main thing to note here, is that the hypotenuse is the largest side of all three sides.
So the equation Arial set up: [tex]9^2+15^2=12^2[/tex] is incorrect, since the 15 would need to be on the right side. This forms the correct equation: [tex]9^2+12^2=15^2[/tex] which then simplifies to: [tex]81 + 144 =225 \implies 225=225[/tex].
Thus a right triangle can be formed using these side lengths. You can of course set up a similar equation to Ariels, where the "c" or hypotenuse is not isolated, but you would have to rearrange the equation so that: [tex]a^2+b^2=c^2\implies b^2=c^2-a^2[/tex] but see how "a squared" is being subtracted from "c squared"? So it's a similar equation to Ariels, but not quite the same, and if she set it up like this, then she would reach the same conclussion
A gardener uses 1/3 of a liter of water to water 2/7 of a garden.
the gardener would need (1 + 1/6) liters of water to water the whole garden.
How much water would the gardener need to water the whole garden?Here we know that the gardener needs 1/3 of a liter of water to water 2/7 of a garden.
Then we have the relation:
1/3 L = 2/7 of a garden.
Now, we want to get a "1 garden" in the right side of the equation, then we can multiply both sides by (7/2), so we get:
(7/2)*(1/3) L = (7/2)*(2/7) of a garden
(7/6)L = 1 garden.
(1 + 1/6) L = 1 garden
This means that the gardener would need (1 + 1/6) liters of water to water the whole garden.
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Find the area of the surface the part of the plane y=x^2 y^2 cylinder x^2 z^2=16
The area of the surface is 144.708
The equation of the given surface is,
z=g(x,y)=xy
Solving the partial derivatives,
∂g∂x=y,∂g∂y=x
Substituting to the formula
S=∬√1+( ∂g∂x)2+(∂g∂y)2dA
Thus,
S=∬√1+(y)2+(x)2dAS=∬√1+x2+y2dA
The region in the XY-plane is defined by the intervals 0≤θ≤2π,0≤r≤4
Converting the integral into polar coordinates,
S=∫2π0∫40√1+r2rdrdθ
Integrating with respect to r
S=∫2π0[13(1+r2)32]40dθ
S=∫2π0(17√173−13)dθ
Integrating with respect to θ
S=(17√173−13)[θ]2π0
S≈144.708
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A ride at a park has room for 8 children to ride at once.
How many rides are needed to allow 45 children a turn?
Answer:
6
Step-by-step explanation:
45/8=5.625
Since there are .625 children left, we need to add another ride to fill in what's left.
The graph of a quadratic function with vertex (1,-3) is shown in the figure below.
Find the domain and the range.
Answer:
Domain: (-∞, ∞), Range: [-3, ∞)
Step-by-step explanation:
Domain is all of the possible x-values in a solution set, in this case since the solution extends infinitely horizontally it would be:
(-∞, ∞)
The range is just all of the possible y-values in a solution set, which in this set starts from -3 and proceeds on infinitely, making the range:
[-3, ∞)
A system of equations is shown:
4x = -3y + 17
3x - 4y = -6
What is the solution to this system of equations? (5 points)
(3,2)
(-3,-2)
(-2,-3)
(2,3)
Answer: (2;3).
Step-by-step explanation:
[tex]\displaystyle\\\left \{ {{4x=-3y+17} \atop {3x-4y=-6}} \right. \ \ \ \ \left \{ {{4x+3y=17\ |*4} \atop {3x-4y=-6\ |*3}} \right. \ \ \ \ \left \{ {{16x+12y=68} \atop {9x-12y=-18}} \right. \\Let's\ sum \ up\ these\ equations:\\25x=50\ |:25\\x=2.\\4*2=-3y+17\\8+3y=17\\3y=9\ |:3\\y=3.[/tex]
Aliana is a certain age, and her mother is six times as old as her. In ten years'
time, her mother will be twice as old as her. Determine Aliana's current age and
her mother's current age.
Answer:
Allana is 2 1/2 and her mother is 15 years old.
Step-by-step explanation:
Let Allana's age be x years then,
her mother's age is 6x.
Using the given information in ten years time we have the equation:-
6x + 10 = 2(x + 10)
6x + 10 = 2x + 20
6x - 2x = 20 - 10
4x = 10
x = 2 1/2
So 6x = 6 * 2 1/2 = 15.
Please help and explain!!!
Answer:
Option A
Step-by-step explanation:
The solution is in the image
Which number line shows the solution to the inequality? y minus 2 less-than negative 5 A number line going from negative 8 to positive 2. An open circle is at negative 3. Everything to the left of the circle is shaded. A number line going from negative 8 to positive 2. A closed circle is at negative 3. Everything to the left of the circle is shaded. A number line going from negative 8 to positive 2. An open circle is at negative 3. Everything to the right of the circle is shaded. A number line going from negative 8 to positive 2. An open circle is at negative 7. Everything to the left of the circle is shaded.
The number line which represents the solution bro the inequality y -2 < -5 is;
An open circle is at negative 3. Everything to the left of the circle is shaded.
First, we must solve the inequality so that it looks it's simplest form in order to be represented on a number line.
Solving the inequality is as follows;
y - 2 < -5
y < -5 +2
y < -3.
In essence, representation of the inequality, y < -3 on the number line is as follows;
An open circle is at negative 3. Everything to the left of the circle is shaded.
PS: It is only a closed circle if the inequality includes an equal to sign.
