The Pascal triangle terms and binomial are related by the binomial theorem fomrula:
The row of the Pascal triangle is the same as
where n is the same as the row of the Pascal triangle.
For example,
.
is the same as the 2nd row of the Pascal triangle.
.
Also know the leading coefficient and last term degree will be the nth row of the Pascal triangle.
The first term, a will be 1 less than the previous terms and the second term, b will be 1 more than the previous terms. For example, look at
.
Another example is
.
The third row of Pascal Triangle
.
.
Pascal's triangle provides a systematic way to determine the coefficients of binomial expansions, making it a valuable tool in algebra and combinatorics.
Binomial expansions are directly related to Pascal's triangle.
Pascal's triangle is a triangular arrangement of numbers in which each number is the sum of the two numbers directly above it. The first and last numbers in each row of Pascal's triangle are always 1.
The coefficients of the terms in a binomial expansion correspond to the numbers in Pascal's triangle. Each row of Pascal's triangle represents the coefficients of the corresponding power of the binomial.
For example, in the expansion of (a + b)², the coefficients are 1, 2, and 1. These coefficients can be found in the third row of Pascal's triangle: 1, 2, 1. Similarly, in the expansion of (a + b)³, the coefficients are 1, 3, 3, and 1, which can be found in the fourth row of Pascal's triangle: 1, 3, 3, 1.
In summary, Pascal's triangle provides a systematic way to determine the coefficients of binomial expansions, making it a valuable tool in algebra and combinatorics.
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A plank 6m long leans against a vertical wall so that the foot of the plank is 4m away from the wall. A lizard climbs 2m up the plank. Calculate the horizontal distance between the lizard and the wall.
Answer: [tex]\dfrac{8}{3}\ m[/tex]
Step-by-step explanation:
Given
Length of the plank is [tex]6\ m[/tex]
Foot of the flank is [tex]4\ m[/tex] away from the wall
Lizard climbs 2 m up the wall
from the figure, the two triangles are similar
[tex]\therefore \dfrac{2}{6}=\dfrac{x}{4}\\\\\Rightarrow x=4\times \dfrac{2}{6}\\\\\Rightarrow x=\dfrac{4}{3}\ m[/tex]
So, the distance from the wall is
[tex]\Rightarrow 4-x\\\\\Rightarrow 4-\dfrac{4}{3}\\\\\Rightarrow \dfrac{8}{3}\ m[/tex]
HELP PLSSS! .
Skylar buys ice cream and lemons at the store.
She
pays
a total of $19.67.
She
pays $5.35 for the ice cream.
She buys 4 bags of lemons that each cost the same amount.
.
Which tape diagram could be used to represent the context if b.
represents
how much each bag of lemons costs?
Answer:
It is the diagram A
Find the volume of a rectangular solid that
is 9 ft. by 12 ft. by 8 ft.
Help please
Answer:
864 ft³ is the volume
Step-by-step explanation:
Volume = width · height · length
= 12 · 9 · 8
= 864
Answer:
Step-by-step explanation:
Volume of rectangular solid = length * width * height
= 9 * 12 * 8
= 864 ft³
Watch help video
Which value of x satisfies the equation
Answer:
-1
Step-by-step explanation:
1.25(x+1.5)=0.625
(x+1.5)=0.5
x=-1
#What is the value of the discriminant for the quadratic equation –3 = –x2 + 2x?
Discriminant = b2 – 4ac
–8
4
8
16
Answer:
16
Step-by-step explanation:
the quadratic equation –3 = –x2 + 2x can be changed into :
x²-2x-3= 0
a=1, b= -2 , and c = -3
so, the discriminant = (-2)²-4(1)(-3)
= 4 + 12 = 16
Given HM = 24 and mHK = 66 , find x and y
Answer:
x=24,y=33
hopefully this answer can help you to answer the next question
Explain what it means for a solution of an equation to be extraneous.
Answer:
Extraneous Solutions An extraneous solution is a root of a transformed equation that is not a root of the original equation because it was excluded from the domain of the original equation. Example 1: Solve for x, 1 x − 2 + 1 x + 2 = 4 (x − 2) (x + 2).
Jenny asked 12 students how many courses they have taken so far at her college. Here is the list of answers.
16, 7, 21, 14, 9, 11, 22, 19, 17, 12, 5, 25
What is the percentage of these students who have taken fewer than 21 courses?
Answer: 75%
Step-by-step explanation:
Only 9 students have taken fewer than 21 courses
9/12 = 0.75 or 75%
Can someone please help? I'm not doing well.
Step-by-step explanation:
am not seeing it is not clear capture another video
Andrew has a coupon for 20 % off of the price of any CD. If he purchases a CD originally priced at $16, what will the discounted price be?
Solve the above problem by using decimals.
Answer:
If you want to turn a percentage into a fraction divide by 100:
20%/100 = 0.2
0.2*16 = 3.2$
and if you want to turn a decimal into a fraction multiply by 100.
His discounted price will be 3.2$.
Have a nice day! :-)
and if you have questions please ask!
Percentage is the part of a whole given as a fraction of the hundred. The discounted price of the CD is $12.8.
Given information-
Andrew has a coupon for 20 % off of the price of any CD.
The price of the CD purchased by the Andrew is $16.
PercentagePercentage is the part of a whole given as a fraction of the hundred.
As the Andrew has a coupon of 20 percent, the 20 percent will be off whatever he buys. Now the discount price for the $16 priced CD is equal to the 20 percent of the 16. Thus the discount is,
[tex]d=\dfrac{16\times20}{100} \\ [/tex]
[tex]d=3.2[/tex]
Thus the discount on the CD is 3.2.
Now the discounted price of the CD is equal to the difference of the price of the CD and the discount. Thus the discounted price of the CD is,
[tex]P=16-3.2[/tex]
[tex]P=12.8[/tex]
Thus the discounted price of the CD is $12.8.
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need help asap pls !
Answer:
L={∅}
Step-by-step explanation:
Match the answers……………..
9 in 8956 = 900
9 in 95675 = 90000
9 = 9 in 124569
9 in 68795 = 90
90000 = 9 in 2549652.........
hope it helps...
Please help me. if I don’t complete my course I’m getting sent back to my parents house, and I need to complete this course
This is similar polygons, and the lessons I get don’t help
DJVJNNSMKAJABJSHSNSB
see attached photo, please help asap!!!
Answer:
3.398
Step-by-step explanation:
log(10) 2500 = log(10) (5^2*10^2) = log(10) 5^2 + log(10) 10^2 = 2*log(10) 5 + 2=3.398
Please answer this, I need help asap
Answer:
Final solution is ---> 17
See attached picture for step-by-step explanation
Answer:
17
Step-by-step explanation:
A fraction bar is like parentheses
Complete the items above and below it first
43-1
------- + 10
4+2
42
----- + 10
6
Then divide
7+10
Then add
17
x+1=0
Giải hộ em với
Answer:
x = -1
Step-by-step explanation:
-Subtract 1 from both sides of the equation
x + 1 = 0
x + 1 - 1 = 0 - 1
-simplify
x = -1
Complete the table for the given rule.
Rule: y=2x
Answer:
1. y=18
2. x=5
3. y=2
Step-by-step explanation:
y=2x
1.
As you have x, substitute (replace) x for 9 to find y.
2×9=18 (We times 2 by 9 because, in algebra, when a number is next to a letter, it means to times).
So, y=18
2.
As we know that y=10, do the same thing and substitute.
10=2x
Isolate x by dividing:
10÷2=5
So, x=5
3.
Substitute x:
2×1=2
So, y=2
Hope this helps :)
What is the reminder? Help
Answer:
6
Step-by-step explanations:
The remainder is what is left over after dividing whatever from d. So if you add one to d, then the remainder would increase from 5 to 6.
Hope this helps.
Find the value of the variable(s). (9x + 12) 3x (4y - 10)
Answer:
2.07648468018
Step-by-step explanation:
Item 1 Which fraction is equivalent to 3.47? 3_23/50 3_47/100 3_12/25 I don't know.
Answer: 3 and 47/100
Step-by-step explanation:
If a state issued license plates using the scheme of two letters followed by four digits, how many plates could it issue?
Answer:
Combining these results, it follows that there are 676 x 1000 = 676,000 different license plates possible.
Write the equivalent expression for the following radical expressions. Simplify numerical expressions
when possible. (1 point)
please help asap
Answer:
Step-by-step explanation:
8) [tex]\sqrt{x^{3}} = x\sqrt{x}[/tex]
9) [tex]\sqrt[3]{3^{6} } = 3^{2} = 9[/tex]
10) [tex](x^{\frac{1}{2} } )^{9} = x^{2} \sqrt{x}[/tex]
11) [tex]7^{2} = 49[/tex]
12) [tex]27^{\frac{2}{3} } = \sqrt[3]{27} ^{2} = 9[/tex]
13) 9x[tex]\sqrt[3]{3xy^{2} }[/tex]
4) If x/y = 7/3 then find the value of
[tex]3x ^{2} + 2y^{2} \3x^{2} - 2y {}^{2} [/tex]
Answer:
1011
Step-by-step explanation:
3x^2 + 2y^2 x^2 - 2y^2
3(7)^2 + 2(3)^2 3^2 - 2(7)^2
3(7^2 + 2 x 3 x 7^2 - 2 x 3)
3(7^2 +6 x 7^2 - 6)
3(7 x 7^2 - 6)
3(7^3 - 6)
3(343 - 6)
3(337) = 1011
20) solve:
[tex] {8}^{2} + 2 = [/tex]
21) solve:
[tex]4(2x + 5y = [/tex]
22) simplify the expression
[tex]4( {2}^{2} + 30) - 4 = [/tex]
Andrew is an avid archer. He launches an arrow that takes a parabolic path.
The equation of the height of the arrow with respect to time is
y = -4.9x2 + 48x, where y is the height of the arrow in meters above
Andrew's bow and x is the time in seconds since Andrew shot the arrow.
Find how long it takes the arrow to come back to a height even with his bow
height.
Answer:
9.7959 sec
Step-by-step explanation:
For the arrow to reach the same height as the bow again, - 4.9x^2+48x=0, 48=4.9x, x=48/4.9=9.7959
The time arrow take to come back to a height even with his bow height is 9.79 seconds.
We have an equation of the height of the arrow with respect to time -[tex]y = -4.9x^{2} +48x[/tex] where y is the height of the arrow in meters above Andrew's bow and x is the time in seconds since Andrew shot the arrow.
We have to find out - how long it takes the arrow to come back to a height even with his bow height.
The motion of arrow in the above situation is an example of which type of motion?It is an example of two - dimensional Projectile motion.
We have the function that depicts the variation of height of the arrow with respect to time given by -
[tex]y=-4.9x^{2} +48x[/tex]
To find the time taken by the arrow to come to a height even with his bow height, we should equate y = 0.
[tex]y=-4.9x^{2} +48x=0\\-4.9x(x-9.79)=0\\-4.9x=0\;\;\;and\;\;\;x-9.79=0\\x =0\;\;\;and\;\;\;x=9.79[/tex]
Time cannot be 0, hence the time arrow take to come back to a height even with his bow height is 9.79 seconds.
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Answer hurry which one 1 2 3 or 4?
Answer:
Option: 1Step-by-step explanation:
Let,
Footprints that Adele will find be = f
So, Footprints that Yumiko finds = 2f = f + f
Since,
As given that,
Yumiko finds 16 Footprints so it will be,
2f = f + f = 16
And,
f = 16 / 2
= 8
Which is half of Footprints found by Yumiko.
So the bar model 1 represents it correct.
Answer:
option 1
Step-by-step explanation:
i did the quiz
please help ! pick two answers
Answer:
B and C
Step-by-step explanation:
25% of 60 = 15
B. 25/100*60 = 15
C. 0.25*60 = 15
A person walks away from a pulley pulling a rope slung over it. The rope is being held at a height 10 feet below the pulley. Suppose that the weight at the opposite end of the rope is rising at 4 feet per second. At what rate is the person walking when s/he is 20 feet from being directly under the pulley
The image of this question is missing and so i have attached it.
Answer:
dd/dt = 4.47 ft/s
Step-by-step explanation:
From the image attached, let's denote the following;
d = horizontal distance beneath pulley
h = height of pulley
l = diagonal from the pulley to the head of the person
v = velocity of rope rising
Using pythagoras theorem;
l² = d² + h²
Differentiating with respect to time and considering h = c^(te) gives;
2l(dl/dt) = 2d(dd/dt)
We are given;
d = 20 ft
h = 10 ft
v = 4 ft/s
We know that velocity in this case is change in diagonal distance with time. Thus;
v = dl/dt = 4 ft/s
From earlier, we saw that;
2l(dl/dt) = 2d(dd/dt)
Thus, reducing it gives
(dl/dt)(l/d) = dd/dt
Now, l² = d² + h²
l = √(d² + h²)
Also, v = dl/dt = 4
Thus;
4(√(d² + h²))/d = dd/dt
4(√(20² + 10²))/20 = dd/dt
dd/dt = 4.47 ft/s
Triangle ABC is a right triangle.
Triangle A B C. Angle A is x degrees, B is 90 degrees, C is (x minus 10) degrees. The exterior angle to angle C is (2 x + 40) degrees.
Which equations can be used to find the value of x? Check all that apply.
x + 90 + (x minus 10) = 180
x + 90 + (2 x + 40) = 180
2 x + 80 = 180
x + 90 = 2 x + 40
(x minus 10) + 90 = 2 x + 40
Answer:
x + 90 + (x minus 10)= 180° can be used to find the value of x.
Answer:
A) X+90+ (x-10) - 180
C) 2x+80 = 180
D) x+90 = 2x+40
Step-by-step explanation:
