Which set of coordinates satisfies the equations 3x - 2y =15 and 4x

If n is a positive integer, how many 5-tuples of integers from 1 through n can be formed in which the elements of the 5-tuple ar

Oksana_A [137]

Answer:

$n + 4 {n \choose 2} + 6 {n \choose 3} + 4 {n \choose 4} + {n \choose 5}$

Step-by-step explanation:

Lets divide it in cases, then sum everything

Case (1): All 5 numbers are different

In this case, the problem is reduced to count the number of subsets of cardinality 5 from a set of cardinality n. The order doesnt matter because once we have two different sets, we can order them descendently, and we obtain two different 5-tuples in decreasing order.

The total cardinality of this case therefore is the Combinatorial number of n with 5, in other words, the total amount of possibilities to pick 5 elements from a set of n.

${n \choose 5 } = \frac{n!}{5!(n-5)!}$

Case (2): 4 numbers are different

We start this case similarly to the previous one, we count how many subsets of 4 elements we can form from a set of n elements. The answer is the combinatorial number of n with 4 ${n \choose 4} .$

We still have to localize the other element, that forcibly, is one of the four chosen. Therefore, the total amount of possibilities for this case is multiplied by those 4 options.

The total cardinality of this case is $4 * {n \choose 4} .$

Case (3): 3 numbers are different

As we did before, we pick 3 elements from a set of n. The amount of possibilities is ${n \choose 3} .$

Then, we need to define the other 2 numbers. They can be the same number, in which case we have 3 possibilities, or they can be 2 different ones, in which case we have ${3 \choose 2 } = 3$ possibilities. Therefore, we have a total of 6 possibilities to define the other 2 numbers. That multiplies by 6 the total of cases for this part, giving a total of $6 * {n \choose 3}$

Case (4): 2 numbers are different

We pick 2 numbers from a set of n, with a total of ${n \choose 2}$ possibilities. We have 4 options to define the other 3 numbers, they can all three of them be equal to the biggest number, there can be 2 equal to the biggest number and 1 to the smallest one, there can be 1 equal to the biggest number and 2 to the smallest one, and they can all three of them be equal to the smallest number.

The total amount of possibilities for this case is

$4 * {n \choose 2}$

Case (5): All numbers are the same

This is easy, he have as many possibilities as numbers the set has. In other words, n

Conclussion

By summing over all 5 cases, the total amount of possibilities to form 5-tuples of integers from 1 through n is

$n + 4 {n \choose 2} + 6 {n \choose 3} + 4 {n \choose 4} + {n \choose 5}$

I hope that works for you!

4 0

3 years ago

Select two ratios that are equivalent to 4:18

fredd [130]

36:8, 9:2, and 72: 16

Step-by-step explanation:

two ratios that are equivalent to 4:18

36:8, 9:2, and 72: 16

would all be equivalent. In order to find ratios that are equivalent to 18:4, you have to multiply both numbers by the same factor. For

8 0

2 years ago

What is the surface area of the rectangular prism below 7, 7, 14

Julli [10]

Answer:

Surface Area A = 490 cm²

Step-by-step explanation:

Surface area of a prism A = 2(w l +h l + h w)

where l = length

w = width

h = height

l = 7 cm; w = 7 cm ; h = 14 cm

Surface Area = 2 { ( 7 × 7) + ( 14 × 7) + ( 14 × 7)}

A = 2 { 49 + 98 + 98}

A = 2 × 245

A = 490 cm²

8 0

2 years ago

(((QUICK QUESTION-WILL GIVE BRAINLYEST)))

svetoff [14.1K]

Given:

Point (7,12) is rotated 1260° counterclockwise about the origin.

To find:

The x-coordinate of the point after this rotation.

Solution:

If a point is rotated 360 degrees then its coordinates remains unchanged.

If a point is rotated 180 counterclockwise about the origin degrees, then

$(x,y)\to (-x,-y)$

We know that,

$1260^\circ=1080^\circ+180^\circ$

$1260^\circ=3\times 360^\circ+180^\circ$

After $3\times 360^\circ$ rotation the coordinates of points remains same, i.e., (7,12). So, after that (7,12) is rotated 180° counterclockwise about the origin.

$(7,12)\to (-7,-12)$

The point (7,12) becomes (-7,-12) after rotation of 1260° counterclockwise about the origin.

Therefore, the x-coordinate of the required point is -7.

3 0

2 years ago

PLEASE HELP!!<br><br><br>11 (5/9 + 2/3) as a fraction

kozerog [31]

2/3 x 3/3 = 4/9
4/9+5/9=9/9=1
1+11= 12
The answer is 12

3 0

2 years ago

Which set of coordinates satisfies the equations 3x - 2y =15 and 4x - y = 20