Consider a stack $S$ and a queue $Q$. Both of them are initially empty and have the capacity to store ten elements each. The elements $1, 2, 3, 4,$ and $5$ arrive one by one, in that order. When an element arrives, it is assigned either to $S$ (pushed on $S$) or to $Q$ (enqueued to $Q$). Once all the five elements are stored, the output is generated in two steps. First, stack $S$ is emptied by popping all elements. Then queue $Q$ is emptied by dequeueing all elements. The output obtained by following this process is $4\ 3\ 1\ 2\ 5$ . Given the output, the objective is to predict whether an element was assigned to $S$ or $Q$. Which of the following options is/are possible valid assignment(s) of the elements? *Note:* In the options, the notation $xS$ denotes that element $x$ was assigned to $S$ and $yQ$ denotes that element $y$ was assigned to $Q$.

The elements arrive in the order $1, 2, 3, 4, 5$.

When all elements have been assigned:

1. The stack $S$ is popped completely (LIFO order: reverse of arrival order in $S$).

2. The queue $Q$ is dequeued completely (FIFO order: same as arrival order in $Q$).

The combined output produced is $4, 3, 1, 2, 5$.

Let us test each option:

• Option (A): $1S, 2Q, 3S, 4S, 5Q$

  • Elements pushed to $S$ in arrival order: $1, 3, 4$. Popping $S$ gives: $4, 3, 1$.

  • Elements enqueued to $Q$ in arrival order: $2, 5$. Dequeueing $Q$ gives: $2, 5$.

  • Combined output: $4, 3, 1, 2, 5$. (Valid)

• Option (B): $1Q, 2Q, 3S, 4S, 5Q$

  • Elements pushed to $S$ in arrival order: $3, 4$. Popping $S$ gives: $4, 3$.

  • Elements enqueued to $Q$ in arrival order: $1, 2, 5$. Dequeueing $Q$ gives: $1, 2, 5$.

  • Combined output: $4, 3, 1, 2, 5$. (Valid)

• Option (C): $1Q, 2Q, 3Q, 4S, 5S$

  • Elements pushed to $S$: $4, 5$. Popping $S$ gives: $5, 4 \neq 4, 3, \dots$ (Invalid)

• Option (D): $1S, 2S, 3S, 4Q, 5Q$

  • Elements pushed to $S$: $1, 2, 3$. Popping $S$ gives: $3, 2, 1 \neq 4, 3, \dots$ (Invalid)

Thus, options (A) and (B) are correct.