No Of ways of arranging plates belonging to two different sets?

Question

There are $n$ triangular plates on which numbers from $1$ to $n$ are written and there are $m$ circular plates on each of them number from $1$ to $m$ are written. We have to find the total no ways of arranging these taken all at time such that triangular and circular plates are in order respectively.

E.g., there are $3$ circular plates named as $c_1, c_2, c_3$ also there are $4$ triangular plates named as $t_1, t_2, t_3, t_4$ One of the possible arrangements of plates is

$t_1~c_1~c_2~t_2~c_3~t_3~t_4$

$t_1, t_2, t_3, t_4$ are in order; $c_1, c_2, c_3$ are in order respectively.

Answer 1

Without loss of generality let $n\ge m$. Then we can order triangular plates first and then place circular plates in-between: $$* \ t_1 \ * \ t_2 \ * t_3 \ * * \ * t_{m-1} \ * \ t_m \ * t_{m+1} \ * \ * \ * t_n$$ $c_1$ can be immediately before or after $t_1$, $c_2$ can be immediately before or after $t_2$ and so on $c_m$ can be immediately before or after $t_m$. So, there are $2^m$ ways to order them.

For example, for your case $n=3$, $m=4$, there are $2^3=8$ ways: $$c_1t_1 \ c_2t_2 \ \color{red}{c_3t_3} \ t_4;\\ c_1t_1 \ c_2t_2 \ \color{red}{t_3c_3} \ t_4;\\ c_1t_1 \ \color{red}{t_2c_2} \ c_3t_3 \ t_4;\\ \color{red}{t_1c_1} \ c_2t_2 \ c_3t_3 \ t_4;\\ c_1t_1 \ \color{red}{t_2c_2 \ t_3c_3} \ t_4;\\ \color{red}{t_1c_1} \ c_2t_2 \ \color{red}{t_3c_3} \ t_4;\\ \color{red}{c_1t_1 \ c_2t_2} \ c_3t_3 \ t_4;\\ \color{red}{c_1t_1 \ c_2t_2 \ t_3c_3} \ t_4;\\ $$

Answer 2

Hint: Since the numbers on the plates of each type must appear in increasing numerical order, an arrangement is completely determined by choosing which $n$ of the $n + m$ positions required for $n$ triangular plates and $m$ circular plates are occupied by the triangular plates.

For example, consider the case of three circular plates and two triangular plates. The ten admissible arrangements are:

$c_1c_2c_3t_1t_2$

$c_1c_2t_1c_3t_2$

$c_1c_2t_1t_2c_3$

$c_1t_1c_2c_3t_2$

$c_1t_1c_2t_2c_3$

$c_1t_1t_2c_2c_3$

$t_1c_1c_2c_3t_2$

$t_1c_1c_2t_2c_3$

$t_1c_1t_2c_2c_3$

$t_1t_2c_1c_2c_3$