Final answer:
In part A, there are 230,230 ways to distribute 30 batteries among eight different types. In part B, there are 53,130 ways to distribute 30 batteries among eight different types, considering the restrictions on A76 and D303 batteries.
Step-by-step explanation:
In part A, to find the number of ways a total inventory of 30 batteries can be distributed among the eight different types, we can use the concept of stars and bars. Since the camera shop can obtain at most ten A76 batteries, we can distribute the remaining 20 batteries among the other seven types. This can be done with 20 stars and 7 bars, representing the batteries and the types, respectively. The number of ways to arrange these stars and bars is given by the formula C(n+k-1,k-1), where n is the number of stars (20) and k is the number of bars (7). Therefore, the number of ways is C(26,6) = 230,230.
In part B, since the store can only obtain 10 A76 batteries and D303 batteries, we can distribute the remaining 20 batteries among the other six types. This can be done with 20 stars and 6 bars, representing the batteries and the types, respectively. The number of ways to arrange these stars and bars is given by the formula C(n+k-1,k-1), where n is the number of stars (20) and k is the number of bars (6). Therefore, the number of ways is C(25,5) = 53,130.