All categories

2 years ago

Prove that C(50,3) + C(50,4) = C(51,4)

Mathematics

1 answer:

valkas [14]2 years ago

5 0

By definition of the binomial coefficient,

$C(n,k) = \dfrac{n!}{k!(n-k)!}$

so we have

$C(50,3) + C(50,4) = \dfrac{50!}{3!47!} + \dfrac{50!}{4!46!} \\\\ = \dfrac{50!}{3!46!} \left(\dfrac1{47} + \dfrac14\right) \\\\ = \dfrac{50!}{3!46!} \times \dfrac{51}{188} \\\\ = \dfrac{51!}{3!46!} \times \dfrac1{4\times47} \\\\ = \dfrac{51!}{4!47!} \\\\ = \dfrac{51!}{4!(51-4)!} = C(51,4)$

as required.

You might be interested in

Prove that the Greatest Integer Function f: R -> R given by f(x) = [x], is neither one-once nor onto, where [x] denotes the g

Karo-lina-s [1.5K]

F: R → R is given by, f(x) = [x]
It is seen that f(1.2) = [1.2] = 1, f(1.9) = [1.9] = 1
So, f(1.2) = f(1.9), but 1.2 ≠ 1.9
f is not one-one

Now, consider 0.7 ε R
It is known that f(x) = [x] is always an integer. Thus, there does not exist any element x ε R such that f(x) = 0.7

So, f is not onto
Hence, the greatest integer function is neither one-one nor onto.

The answer was quite complicated but I hope it will help you.

8 0

3 years ago

• a helicopter hovering at 10 m above the ground is shining a spotlight at a person 30 m away1 . if the person is 2 m tall how l

pshichka [43]

As shown in the figure

Height of helicopter above the ground=30 m

Distance from the base to the point where the person is standing=30 m

height of the person=2 m

Now the person cast the shadow of length x m.

Triangle ABC and triangle EDC are similar.

∵ ∠B= ∠D=90°

∠C is common.

So by AA similarity ΔABC and ΔEDC are similar.

As we know when triangles are similar their sides are proportional.

AB/AD =BC/DC

Let DC=x meter

⇒ $\frac{10}{x}=\frac{30+x}{x}$

⇒5=\frac{30+x}{x}[/tex]

⇒5x=30+x

⇒4x = 30

⇒ x=30/4

⇒ x=7.5 meter

So length of shadow=7.5 meter

2. volume of a sphere of radius r is v (r) = $\frac{4}{3}π r^{3}$

$\frac{\mathrm{d} }{\mathrm{d} r}V$ =4/3π×3 $r^2$

=4π $r^2$

surface area of a sphere of radius r is 4π $r^2$

b) Ratio of derivative of volume of sphere to surface area of sphere= $\frac{4\pi r^2}{4\pi r^2}$

=1 [ incomplete question but you have written few words ]

4 0

3 years ago

PLS HELO IM MARKING BRAINLEIST!!

Musya8 [376]

∠O ≅ ∠U is third statement which is congruent .

What is congruent triangle?

Two figures are said to be "congruent" if they can be placed perfectly over one another. Both of the bread slices are the same size and shape when stacked one on top of the other.
Congruent refers to having precisely the same form and size.
If all three sides of two triangles are the same, they are said to be congruent.
If we have a side, an angle between the sides, and then a second side that is congruent, we know they are congruent. In other words, side, angle, side.

ΔONP and ΔUTV are congruent .

By using ASA Theorem ,

NP ≅ TV

∠P ≅ ∠V

∠O ≅ ∠U

Learn more about congruent

brainly.com/question/22062407

#SPJ13

4 0

1 year ago

Read 2 more answers

What is the value of x in the equation below?

belka [17]

$remember$
$ln(a^b)=bln(a)$
$ln(e)=1$
$so$
$ln(e^x)=xln(e)=x$

$1+2e^{x+1}=9$
$minus \space\ 1 \space\ both \space\ sides$
$2e^{x+1}=8$
$divide \space\ both \space\ sides \space\ by \space\ 2$
$e^{x+1}=4$
$take \space\ ln \space\ of \space\ both \space\ sides$
$ln(e^{x+1})=ln(4)$
$(x+1)ln(e)=ln(4)$
$x+1=ln(4)$
$minus \space\ 1 \space\ from \space\ both \space\ sides$
$x=ln(4)-1$
$answer \space\ is \space\ C=ln(4)-1$

5 0

3 years ago

Read 2 more answers

Use distributive law evaluate

Cerrena [4.2K]

Answer:

The distributive property allows you multiply a sum in parenthesis by multiplying each addend separately, then add the products.

Step-by-step explanation:

How to use the distributive law example.

2(x+4) = 16

To use the distributive law in this example multiply 3 by all terms in the parenthesis. Multiply 2 and x, then 2 and 4 to open the parenthesis.

2x+8=16

That is how you use the distributive law.

To continue solving, subtract 8 from both sides.

2x+8-8=16-8

2x=8

Divide 2 from both sides.

2x/2=8/2

x=4

Hope this helps!

If not, I am sorry.

6 0

2 years ago