Question

From among 36 teachers in a college, one principle, one vice-principal and teacher in charge are to be appointed. In how many ways it can be done?

Answer 1

Here from 36 teachers three people are to be selected for different posts. So use the method of combinations.
Formula is $n{C_r}$= $\dfrac{{n!}}{{r!\left( {n - r} \right)!}}$ where n is total number sand r is to be selected,

Complete step-by-step answer:
It is given that,
There are 36 teachers in a college.
From them one principle, one vice-principal and teacher in charge are to be appointed.
That means we need to fill 3 vacant positions.
Let’s start from the post of principle.
Now there are all 36 teachers who can be appointed to this post.
So there are 36 different ways to fill this post.
Next we will go for a vice-principal post. But now we left with 35 teachers only because one teacher is already appointed as principal.
So there are 35 different ways to appoint a vice-principal.
Now left with only the teacher in charge.
So there are 34 different ways to fill this post.
In total we have $36 \times 35 \times 34 = 42840$ ways for this selection.
We can write it as , $36{C_1} \times 35{C_1} \times 34{C_1}$ ways.
So there are 42840 ways for this selection.

Note: Here we can’t directly write it as $36{C_3}$ because they are not all in the same position.
Once a position is filled others will not get that chance to have that post.