Question 15
Let $x_1,x_2,\ldots,x_{10}$ be a permutation of the numbers $1,2,3,4,5,6,7,8,9,10$. How many different permutations satisfy the following conditions?
$$x_1<x_2<x_3<x_4<x_5,$$
$$x_6<x_7<x_8<x_9<x_{10},$$
$$x_1+x_2+x_3+x_4+x_5<x_6+x_7+x_8+x_9+x_{10}.$$
$$x_1<x_2<x_3<x_4<x_5,$$
$$x_6<x_7<x_8<x_9<x_{10},$$
$$x_1+x_2+x_3+x_4+x_5<x_6+x_7+x_8+x_9+x_{10}.$$