Question

Ann is grouping 38 rocks. She can put them into group of 10 rocks or as single rocks. What are the different ways Ann can group the rocks?

Answer 1

(A) If the rocks are identical, then there are four ways:
38 singles
1 group of 10+28 singles
2 groups of 10+18 singles
3 groups of 10+8 singles

(B) If each rock is different from others then 38 rocks. In groups of 10 or singles.

Case 0: 0 group+38 singles
1 way

Case 1: 1 group+28 singles:
38C10=38!/(10!(28!))=472733756 ways

case 2: 2 groups of 10+18 singles 38C20*20C10=38!/(20!18!)*20!/(10!10!)/2!=3101868540350580 ways

Case 3: 3 groups of 10+8 singles
38C30*30C20*20C10/3!=45243854529553559880 ways

Total
=1+472733756+3101868540350580+45243854529553559880
=45246956398566644217 ways
Sorry that it is not possible to enumerate each of the ways, but just the total number of ways.