Points in space

There are n points, of which no three lie on one line and no four lies on one plane. How many planes can be guided by these points? How many planes are there if there are five times more than the given points?

Correct answer:

r =  10
r₂ =  35

Step-by-step explanation:

$$\begin{gathered} C_3(10)=\left({\displaystyle \genfrac{}{}{0px}{}{10}{3}}\right)={\displaystyle \frac{10!}{3!(10-3)!}}={\displaystyle \frac{10\cdot 9\cdot 8}{3\cdot 2\cdot 1}}=120 \\ n=10 \\ r=\left(\genfrac{}{}{0px}{}{n}{3}\right)=120=10 \end{gathered}$$

$$\begin{gathered} C_3(7)=\left({\displaystyle \genfrac{}{}{0px}{}{7}{3}}\right)={\displaystyle \frac{7!}{3!(7-3)!}}={\displaystyle \frac{7\cdot 6\cdot 5}{3\cdot 2\cdot 1}}=35 \\ 5n=\left(\genfrac{}{}{0px}{}{n}{3}\right) \\ 5n=n\text{ /}3 \\ 5n=n(n-1)(n-2)(n-3)\text{ /}3 \\ 5=(x-1)(x-2)\mathrm{/}6 \\ -0.16666666666667x^2+0.5x+4.667=0 \\ 0.16666666666667x^2-0.5x-4.667=0 \\ a=0.16666666666667;b=-0.5;c=-4.667 \\ D=b^2-4ac=0.5^2-4\cdot 0.16666666666667\cdot (-4.667)=3.3611111111 \\ D>0 \\ {x}_{1,2}={\displaystyle \frac{-b\pm \sqrt{D}}{2a}}={\displaystyle \frac{0.5\pm \sqrt{3.36}}{0.33333333333333}} \\ {x}_{1,2}=1.5\pm 5.5 \\ {x}_1=7 \\ {x}_2=-4 \\ \text{ Factored form of the equation: } \\ 0.16666666666667(x-7)(x+4)=0 \\ {r}_2=\left(\genfrac{}{}{0px}{}{x_1}{3}\right)=35=35 \end{gathered}$$

Our quadratic equation calculator calculates it.

Try another example

Related math problems and questions:

• Lines
  How many lines can be draw with 8 points, if three points lie on one line and the other any three points do not lie on the same line?
• Points in plane
  The plane is given 12 points, 5 of which is located on a straight line. How many different lines could by draw from this points?
• Circles
  How many different circles are determined by 9 points at the plane if 6 of them lie in a straight line?
• Combinations
  If the number of elements increase by 3, it increases the number of combinations of the second class of these elements 5 times. How many are the elements?
• Combinations
  How many elements can form six times more combinations fourth class than combination of the second class?
• Curve and line
  The equation of a curve C is y=2x² -8x+9, and the equation of a line L is x+ y=3 (1) Find the x coordinates of the points of intersection of L and C. (2) Show that one of these points is also the stationary point of C?
• Combinations 6
  6 purses 9 flaps 12 straps Every combination must include 1 purse, 1 flap, and 1 strap. How many are possible combinations?
• Combinations
  From how many elements we can create 990 combinations 2nd class without repeating?
• Twelve flowers
  A florist has roses, tulips, daffodils, and carnations to use in flower arrangements. If she were to make an arrangement using 12 flowers, how many different combinations of these 4 types of flowers would be possible?
• Five-digit numbers
  How many different five-digit numbers can be created from the numbers 2,3,5 if the number 2 appears in the number twice and the number 5 also twice?
• Cinema
  How many ways can be divided 11 free tickets to the premiere of "Jáchyme throw it in the machine" between 6 pensioners?
• Lines
  In how many points will intersect 14 different lines, where no two are parallel?
• Fruits
  In the shop sell 4 kinds of fruits. How many ways can we buy three pieces of fruit?
• 2nd class combinations
  From how many elements you can create 4560 combinations of the second class?
• Bouquets
  In the flower shop they sell roses, tulips and daffodils. How many different bouquets of 5 flowers can we made?
• N points on the side
  There is an equilateral triangle A, B, C on each of its inner sides lies N=13 points. Find the number of all triangles whose vertices lie at given points on different sides.
• Line
  Straight-line passing through points A [-3; 22] and B [33; -2]. Determine the total number of points of the line in which both coordinates are positive integers.