Question

an employee of a grocery store is placing an order for soda. there are 6 varieties of soda, and they are sold in cases. each case contains all the same variety of soda. the store will order 15 cases total. (for each question, you should clearly explain how you get the answer.) (20 points) a. how many ways are there to select these 15 cases from 6 varieties? (10 points) b. how many ways are there to select these 15 cases if the employee orders at least 1 of each variety? (10 points)

Answer 1

a. There are 0.0001161048 ways to select 15 cases from 6 varieties. b. There are 504 ways to select 15 cases if the employee orders at least 1 of each variety.

Step-by-step explanation:

a. To select 15 cases from 6 varieties, you can use the concept of combinations. The formula to calculate the number of ways to select r items from a set of n items is nCr = n! / (r! * (n-r)!).

In this case, the number of ways to select 15 cases from 6 varieties is

6C15 = 6! / (15! * (6-15)!)

= 6! / (15! * (-9)!)

= 6! / (15! * 9!)

= (6 * 5 * 4 * 3 * 2 * 1) / (15 * 14 * 13 * 12 * 11 * 10 * 9!)

= 720 / (130 * 12 * 11 * 10 * 9!)

= 720 / (171600 * 9!)

= 720 / (171600 * (9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1))

= 720 / (171600 * 362880)

= 720 / 62270208000

= 1 / 8610

= 0.0001161048

b. To select 15 cases if the employee orders at least 1 of each variety, we can first distribute 1 case of each variety. Then, we have 9 cases left to distribute. Using the concept of combinations, the number of ways to distribute these 9 cases among 6 varieties is 9C6 = 9! / (6! * (9-6)!)

= 9! / (6! * 3!)

= (9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) / (6 * 5 * 4 * 3 * 2 * 1 * 3 * 2 * 1)

= 9 * 8 * 7 / 3 * 2 * 1

= 504