Question

In how many ways first and second rank in Mathematics, first and second rank in Physics, first rank in Chemistry and first rank in English is given away to a class of $30$ students.

Answer 1

First, we have found all the ways that are present in the given question.
Then, we need to find all the possibilities that are available in each case.
Now check whether you have to add the categories or multiply them to get the result of total possibilities.
So, you must multiply all ranks possibilities for each rank to be secured.

Complete step-by-step solution:
In the question we have given that the student Rank $1$ in mathematics, Rank $2$ in mathematics, Rank in $1$ physics, Rank $2$ in physics, Rank $1$ in chemistry, Rank $1$ in English.
Also the total number of students is $30$.
Here we are mentioning all the possible methods,
So, we can write it as the Rank $1$ is given to $1$ student
Here, we have $29$ students left because $1$ student is already assigned to rank $1$.
By listing each category with their possibilities, we get,
Possibilities for First rank in mathematics can be given in $30$ ways
Possibilities for Second rank in mathematics can be given in $29$ ways
Possibilities for First rank in physics can be given in $30$ ways
Possibilities for Second rank in physics can be given in $29$ ways
Possibilities for First rank in chemistry can be given in $30$ ways
Possibilities for First rank in English can be given in $30$ ways
If a student got rank $1$ in mathematics there is no rule to stop him securing rank $1$ in physics.
So, similarly it can be said for any pair of subjects.
So, we must use product rules.
By applying product rule, we get
Total possibilities {\text{ = }}$$$$\left( {{{30 \times 29}}} \right){{ \times }}\left( {{{30 \times 29}}} \right){{ \times 30 \times 30}}
On multiplying the bracket terms we get,
Total possibilities ${\text{ = }}$ ${\text{68,12,10000}}$

$\therefore$ The total number of ways we have ${\text{68,12,10000}}$ ways.

Note: The possible set of outcomes of a random experiment is the sample space of the individual space of that experiment.
The likelihood of occurrence of an event is called probability. The probability of any event lies between 0 and 1.