Question

There are six teachers of whom two are from Science, two from Arts and the remaining two from Commerce. They have to stand in a line so that the two Science teachers, two Arts teachers and two Commerce teachers are together. Find the number of ways in which they can do so:

Answer 1

Hint: Two Science teachers, two arts teachers and two commerce teachers are to be stood together. So consider two Science teachers as one entity, two arts teachers as one entity and two commerce teachers as one entity. Now there will be three entities and arrange these three using permutations say x. And internally the together stood teachers can also change their respective places so find their arrangements say y. Multiply x with y to get the number of ways in which they can do so.

Complete step-by-step answer:
We are given that there are six teachers of whom two are from Science, two from Arts and the remaining two from Commerce. They have to stand in a line so that the two Science teachers, two Arts teachers and two Commerce teachers are together.
We have to find the number of ways they can be arranged in the above said order.
Total no. of teachers = 6.
So consider two science teachers as one entity, two arts teachers as one entity and two commerce teachers as one entity, because they have to stand together.
Now, the total number of entities is 3. And internally two teachers who stood together can change their places. Here we have to follow a specific order that two teachers must be together. So use Permutations.
Total no. of ways that we can do so is
$
  {}^3{P_3} \times {}^2{P_2} \times {}^2{P_2} \times {}^2{P_2} \\
  {}^n{P_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}} \\
   \to {}^3{P_3} = \dfrac{{3!}}{{\left( {3 - 3} \right)!}} = \dfrac{{3!}}{{0!}} \\
  \because 0! = 1 \\
  \therefore {}^3{P_3} = \dfrac{{3!}}{1} = 3 \times 2 \times 1 = 6 \\
  {}^2{P_2} = \dfrac{{2!}}{{\left( {2 - 2} \right)!}} = \dfrac{{2!}}{{0!}} \\
  \therefore {}^2{P_2} = 2 \times 1 = 2 \\
  \therefore No.of ways = 6 \times 2 \times 2 \times 2 = 48 \\
$
In 48 ways, six teachers can be arranged.

Note: A Permutation is arranging the objects in order. Combinations are the way of selecting the objects from a group of objects or collection. When the order of the objects does not matter then it should be considered as Combination and when the order matters then it should be considered as Permutation. Do not confuse using a permutation, when required, instead of a combination and vice-versa.