Question

Camera shop stocks six different types of batteries, one of which is type A7b. Suppose that the camera shop has only twelve A7b batteries but at least 30 of each of the other types. Now, answer the following question - How many ways can a total inventory of 30 batteries be distributed among the six different types?

Answer 1

The number of ways to distribute 30 batteries among the six different types is 33,649.

Explanation:

It is provided that a camera shop stocks six different types of batteries, one of which is type A7b.

Also, the camera shop has only twelve A7b batteries but at least 30 of each of the other types.

Combinations would be used to determine the number of ways to distribute 30 batteries among the six different types. Here repetition is allowed.

`C(n+r-1, r)={n+r-1\choose r}=((n+r-1)!)/(r!(n-1)!)`

The number of A7b batteries is 12.

Then the number of ways to distribute 30 batteries among the six different types is:

`C(n+(r-k)-1, (r-k))={n+(r-k)-1\choose (r-k)}=((n+(r-k)-1)!)/((r-k)!(n-1)!)`

The number of ways is:

`C(n+(r-k)-1, (r-k))={n+(r-k)-1\choose (r-k)}`

`=((n+(r-k)-1)!)/((r-k)!(n-1)!)\\\\=((6+(30-12)-1)!)/((30-12)!* (6-1)!)\\\\=(23!)/(18!* 5!)\\\\=(23* 22* 21* 20* 19* 18!)/(18!* 5!)\\\\=(23* 22* 21* 20* 19)/( 5!)\\\\=33649`

Thus, the number of ways to distribute 30 batteries among the six different types is 33,649.