Question

In how many ways can 7 people be divided into 3 identical groups? ___

Answer 1

People are distinct and hence we wil have to treat this differently. First we will have to list out the distributions for 7 identical objects, and then for each distribution, we will have to figure out the number of ways for distinct people to be distributed.
(7,0,0) - 1 way
(6,1,0) - $\frac{7!}{6!}$ = 7 ways
(5,2,0) - $\frac{7!}{5!2!}$ = 21 ways
(5,1,1) - $\frac{7!}{5!2!}$ = 21 ways
(4,3,0) - $\frac{7!}{4!3!}$ = 35 ways
(4,2,1) - $\frac{7!}{4!2!}$ = 105 ways
(3,3,1) - $\frac{7!}{3!3!2!}$ = 70 ways
(3,2,2) - $\frac{7!}{3!2!2!2!}$ = 105 ways
Thus the total number of ways is 1 + 7 + 21 + 35 + 105 + 70 + 105 = 365 ways