Question

The number of ways in which 9 persons can be divided into three equal groups:
A) 1680
B) 840
C) 560
D) 280

Answer 1

Hint: When we are dividing similar things into equal groups, the total number of ways would be 1. When it involves different things such as people, we will use the combination formula. When we have to arrange them, then we will use the permutation formula.
Combination formula: $\dfrac{{n!}}{{(n - r)!r!}}$ when we have to choose $r$ things from $n$ things.
Permutation formula: $\dfrac{{n!}}{{(n - r)!}}$ when we have to arrange $r$ things from $n$ things.
Using these two formulae, we will try solving this question.

Complete step by step solution:
Total number of people = 9
Number of groups = 3
First we'll pick 3 people from 9 people.
Total number of ways of choosing is: $\dfrac{{9!}}{{(9 - 3)!3!}}$
After further simplification we get: $\dfrac{{9!}}{{(9 - 3)!3!}} = \dfrac{{9!}}{{6!3!}} = 84$
Now let’s pick another set of 3 people from the remaining 6 people.
Total number of ways of choosing is: $\dfrac{{6!}}{{(6 - 3)!3!}}$
After simplifying we get: $\dfrac{{6!}}{{(6 - 3)!3!}} = \dfrac{{6!}}{{3!3!}} = 20$
The number of ways of picking 3 people from 3 people is 1.
Since the three groups are similar, there is no differentiation between them. Therefore, we need to divide it with $3! = 6$
The total number of ways = $\dfrac{{84 \times 20}}{6} = 280$

Therefore, the correct option is D

Note: When the question involves people, we need to remember that they are different things. If the question would have asked about the number of ways they can be arranged, then we will use the permutation formula. Students should be careful while performing the calculations, a simple mistake can give a wrong answer. We should remember to divide with 6 at the end since the groups are similar.