Respuesta :
There are 28 ways for Mary to put the plants on the sills.
We can use the Binomial Theorem to solve this problem.
We have 6 plants and 3 sills, so we have6 + 3 − 1 = 8 total objects.
Since we have 8 objects, we can use the Binomial Theorem to expand [tex](x + y)^{8}[/tex].
The coefficient [tex]x^{6} y^{2}[/tex] will be the number of ways for Mary to put the plants on the sills. We can expand [tex](x + y)^{8}[/tex]
using the Binomial Theorem:
[tex](x + y)^{8}=(\left\ {{8}\atop {0}} \right. )x^{8}+ (\left\ {{8}\atop {1}} \right. )x^{7}y+(\left\ {{8}\atop {2}} \right. )x^{6}y^{2} +(\left\ {{8}\atop {3}} \right. )x^{5}y^{3} +(\left\ {{8}\atop {4}} \right. )x^{4}y^{4}+(\left\ {{8}\atop {5}} \right. )x^{3}y^{5} +(\left\ {{8}\atop {6}} \right. )x^{2}y^{6} +(\left\ {{8}\atop {7}} \right. )x^{1}y^{7} +(\left\ {{8}\atop {8}} \right. )y^{8}[/tex]
Since we are only interested in the coefficient of [tex]x^{6} y^{2}[/tex] , we can ignore all terms that do not have [tex]x^{6}[/tex] and [tex]y^{2}[/tex].
Therefore, we are left with [tex](\left\ {{8}\atop {2}} \right. )x^{6}y^{2}[/tex]
[tex](\left\ {{8}\atop {2}} \right. )=\frac{8!}{2!(8-2)!} =\frac{8!}{2!6!} =\frac{8*7*6!}{2!6!} =28\\[/tex]
Therefore, there are 28 ways for Mary to put the plants on the sills.
Learn more about Binomial theorem here https://brainly.com/question/2584994
#SPJ4
