SMOJ wrote: Let $X={1,2,3,4,5,6,7,8,9,10}$ and $A={1,2,3,4}$. Find the number of $4$-element subsets $Y$ of $X$ such that $10\in Y$ and the intersection of $Y$ and $A$ is not empty. Note that if we let $ Y=\{x,y,z,10\} $, then the total possible values of $Y$ without any restriction is $C(9,3)=84$. Now if the intersection of $Y$ and $A$ is empty then we have that the total possible values for $Y$ is $C(5,3)=10$. Hence we now have that the required number of subsets is $\boxed {74}$.

SMOJ wrote: Let $X={1,2,3,4,5,6,7,8,9,10}$ and $A={1,2,3,4}$. Find the number of $4$-element subsets $Y$ of $X$ such that $10\in Y$ and the intersection of $Y$ and $A$ is not empty. Note that if we let $ Y=\{x,y,z,10\} $, then the total possible values of $Y$ without any restriction is $9^3=729$. Now if the intersection of $Y$ and $A$ is empty then we have that the total possible values for $Y$ is $5^3=125$. Hence we now have that the required number of subsets is $\boxed {804}$.

without any restriction $C(9,3)=84$ if intersection of$Y$ and $A$ is empty we have total posible values for $Y$ is $C(5,3)=10$ $84-10=74$

SMOJ wrote: Let $X={1,2,3,4,5,6,7,8,9,10}$ and $A={1,2,3,4}$. Find the number of $4$-element subsets $Y$ of $X$ such that $10\in Y$ and the intersection of $Y$ and $A$ is not empty. Note that if we let $ Y=\{x,y,z,10\} $, then the total possible values of $Y$ without any restriction is $9^3=729$. Now if the intersection of $Y$ and $A$ is empty then we have that the total possible values for $Y$ is $5^3=125$. Hence we now have that the required number of subsets is $\boxed {804}$.

without any restriction $C(9,3)=84$ if intersection of$Y$ and $A$ is empty we have total posible values for $Y$ is $C(5,3)=10$ $84-10=74$