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Mother has $$12$$ identical peaches. She wants to place them into $$3$$ identical baskets. How many ways can she place the peaches if the number of peaches in each basket cannot be zero and must be different from one other? 
Overseas Competition->Knowledge Point->Counting Modules->Questions Involving Enumeration->Enumeration
Order does not matter as peaches and baskets are identical. But we need to have different number of peaches in each basket and none of the baskets can be empty.  $$12=1+2+9$$  $$12=1+3+8$$  $$12=1+4+7$$  $$12=1+5+6$$  $$12=2+3+7$$  $$12=2+4+6$$  $$12=3+4+5$$  There are $$7$$ ways she can place the peaches. 