Question

In how many different ways may 12 things, 4 each of three varieties be distributed equally among two persons ? Things of the same variety are assumed to be identical.

Answer 1

If we choose thing for one person automatically the left out things will go to the second person, hence we find out the number of ways of distributing $6$ things to one person.

The possible cases are:

1) Selecting $4$ from one variety and two from other

Number of ways for choosing first variety $=3$

Number of ways of choosing second variety $=2$

Hence the total number of ways $=6$

2) Selecting $3$ from one variety and $3$ from other

Number of ways for choosing two varieties from three $={}^3{C}_2$

Hence the total number of ways $=3$

3) Selecting $4$ from one variety and one from other and one from third

Number of ways for choosing first variety $=3$

Number of ways of choosing second variety $=2$

Number of ways of choosing third variety $=1$

Hence the total number of ways $=6$

4) Selecting $3$ from one variety and $2$ from other and $1$ from third

Number of ways for choosing first variety $=3$

Number of ways of choosing second variety $=2$

Number of ways of choosing third variety $=1$

Hence the total number of ways $=6$

5) Selecting $2$ from each variety

Number of ways of selecting $3$ varieties $={}^3{C}_1$

Hence the total number of ways are $6+3+6+6+1=22$