P n C....open to all

In how many ways can n things be selected out of 3n things of which n are alike of one kind, n are alike of 2nd kind and rest are different?

those beads can be arranged in the three following ways. 1st- all the three red beads are kept separately.2nd-two are kept together and 1 is separate.3rd- three are together.There are 6n spaces between blue beads.Fr d 1st case the no. of selecn is 1/2 6nC3 as clckws and anticlckws are not different. fr d 2nd case no. of selecn in 2*1/2*6nC2 (2 2gether are taken as 1 unit) . 2 is multiplied as 2 beads 2geder & 1 sparate are diffrnt . fr d 3rd case no. of ways is 1/2*6nC1. so d ans will be sum total of dis three.

@ Post # 2.

It can be thought like this.

We have n alike of one kind, n alike of other kind and n different.

So total = 3n. Now since we have to select any n , it could possibly consist of :

n completely different,
n-1 n completely different and the rest 1 from either of the alike groups,
n-2 n completely different and the rest 2 from either of the alike groups..and so on..

I may be incorrect somewhere but the concept is same and after a bit thinking you should reach the result.