How many permutations of $A_1, A_2, A_3, A_4, B_1, B_2, B_3$ have the $A$'s and $B$'s in ascending order. 
How many permutations of $A_1, A_2, A_3, A_4, B_1, B_2, B_3$ have the $A$'s in ascending order and $B$'s in ascending order. (i.e, $A_1, A_2, B_1, A_3, B_2, B_3, A_4 $). Can the solution be extended?($A_1,\dots,A_n,B_1,\dots,B_m,C_1,\dots, C_s$)

 A: See all the elements as if all were dots $\bullet$. Then , you select $3$ of them, and they will be the $A$'s of your arrangement. Since your $A$'s must be in ascending order, you must put them in order. The rest dots willbe the $B$'s in ascending order. Example:
$$ \bullet\bullet\bullet\bullet\bullet\bullet\bullet$$
I select the ones with stars
$$ \bullet\star\star\bullet\star\star\bullet$$
Then the arrangement is
$$B_1,A_1,A_2,B_2,A_3,A_4,B_3$$
The number of arrangements will be
$$\binom{7}{4}=\binom{7}{3}=\frac{7\cdot 6\cdot 5}{3\cdot 2\cdot 1}= 7\cdot 5= 35$$
For the case of $A_1,\dots,A_n,B_1,\dots B_s$, you can do the same: Consider all elements as points, and then select $n$ elements, that will be your $A$'s. Then order them is ascending order, and put the $A$'s. The rest of elements are $B$'s so put them in ascending order too and you will have all the arrangements in this way. The number of arrangements will be
$$\binom{n+s}{n}=\binom{n+s}{s}$$
A: Order the As first: $A_1,A_2,A_3,A_4$.
Now, $B_2$ can be placed at any one of $5$ slots.
When $B_2$ is placed at slot #$n$:


*

*$B_1$ can be placed at any one of $n$ slots

*$B_3$ can be placed at any one of $6-n$ slots


Hence the total number of ways to order the Bs is $\sum\limits_{n=1}^{5}n(6-n)=35$.
