Combination with Repetition: 100 bikes, 4 warehouses question A bike collector has 100 indistinguishable bikes. How many ways can the bikes be stored in four warehouses if warehouse 1 gets at least 10 bikes and warehouse 2 gets at least 10 but at most 20 bikes ?

As a variant on the solution of Benson Lin, and basically as suggested in comments by Doug M and csts, start with only the constraint of at least 10 bicycles in warehouse 1 and 2. This gives you $$ \binom{100-10-10+3}{3} = \binom{83}{3} $$ possibilities. Now count the possibilities when warehouse 1 gets at least 10, and warehouse 2 gets at least 21. The number is $$ \binom{100-10-21+3}{3} = \binom{72}{3}. $$ Subtract the first from the second to get $$ 91\,881 - 59\,640 = 32\,241. $$