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?

Final Answer:

r =  10
r2 =  35

Step-by-step explanation:

C3​(10)=(310​)=3!(10−3)!10!​=3⋅2⋅110⋅9⋅8​=120 n=10  r=(3n​)=120=10
C3​(7)=(37​)=3!(7−3)!7!​=3⋅2⋅17⋅6⋅5​=35 5n = (3n​) 5n = n /3 5n = n(n−1)(n−2)(n−3) /3  5=(x−1)(x−2)/6  −0.16666666666667x2+0.5x+4.667=0 0.16666666666667x2−0.5x−4.667=0  a=0.166667;b=−0.5;c=−4.667 D=b2−4ac=0.52−4⋅0.166667⋅(−4.667)=3.3611111111 D>0  x1,2​=2a−b±D​​=0.3333330.5±3.36​​ x1,2​=1.5±5.5 x1​=7 x2​=−4  r2​=(3x1​​)=35

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You need to know the following knowledge to solve this word math problem:

geometrycombinatoricsalgebrabasic operations and conceptsGrade of the word problem

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