Please help A camera shop stocks eight different types of batteries, one of which is type A76. Assume there are at least 32 batteries of each type.(a) How many ways can a total inventory of 32 batteries be distributed among the eight different types?Following the model of Example 9.6.2, represent each way of distributing the inventory as a string oftypes of batteries corresponding to the spaces around the vertical bars and the batteries corresponding to the crosses. It follows that the number of ways the totalinventory can be distributed isvertical bars andcrosses, with the(b) How many ways can a total inventory of 32 batteries be distributed among the eight different types if the inventory must include at least four A76 batteries?(c) How many ways can a total inventory of 32 batteries be distributed among the eight different types if the inventory includes at most three A76 batteries? Example 9.6.2Selecting 15 Cans of Soft Drinks of Five Different TypesA person giving a party wants to set out 15 assorted cans of soft drinks for his guests. Heshops at a store that sells five different types of soft drinks.a. How many different selections of cans of 15 soft drinks can he make?b. If root beer is one of the types of soft drink, how many different selections include atleast six cans of root beer?C. If the store has only five cans of root beer but at least 15 cans of each other type ofsoft drink, how many different selections are there?ors nod2wank orSolutiona. Think of the five different types of soft drinks as the n categories and the 15 cans ofsoft drinks to be chosen as ther objects (so n = 5 and r = 15). Each selection of cansof soft drinks is represented by a string of 5-1 = 4 vertical bars (to separate thecategories of soft drinks) and 15 crosses (to represent the cans selected). For instance,the stringodmun odi oosx x × | ×X x x × × ||× × × | × Xrepresents a selection of three cans of soft drinks of type 1, seven of type 2, none of) ted Isor type 3, three of type 4, and two of type 5. The total number of selections of 15 cans ofto odo of soft drinks of the five types is the number of strings of 19 symbols, 5-1= 4 of them|edand 15 of them X:to 1odr(*)-6 219.18 17 16 15!15+5-119!(19)15151 -4-3-2-1= 3,876.15!·(19 – 15)!15b. If at least six cans of root beer are to be included in the selection, you can imaginechoosing six such cans first and then choosing nine additional cans. The choice of thenine additional cans can be represented as a string of 9 ×'s and 4 's. For example, ifroot beer is type 1, then the string X × X || X X |X X X X | represents a selectionof three cans of root beer (in addition to the six chosen initially), none of type 2. twoof type 3, four of type 4, and none of type 5. Thus the total number of selections of15 cans of soft drinks of the five types, including at least six cans of root beer, is thenumber of strings of 13 symbols, 4 (=5 – 1) of them | and 9 of them ×:9+413 12-11 10-9!91 -4-3-2-113!139!-(13 – 9)!9= 715.

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A camera shop stocks eight different types of batteries, one of which is type A76. Assume there are at least 32 batteries of each type. (a) How many ways can a total inventory of 32 batteries be distributed among the eight different types? Following the model of Example 9.6.2, represent each way of distributing the inventory as a string of types of batteries corresponding to the spaces around the vertical bars and the batteries corresponding to the crosses. It follows that the number of ways the total inventory can be distributed is vertical bars and crosses, with the (b) How many ways can a total inventory of 32 batteries be distributed among the eight different types if the inventory must include at least four A76 batteries? (c) How many ways can a total inventory of 32 batteries be distributed among the eight different types if the inventory includes at most three A76 batteries?