Question

A store is selling 5 types of hard candies: cherry, strawberry, orange, lemon and pineapple. How many ways are there to choose:(a) 32 candies? (b) 32 candies with at least a piece of each flavor? (c) 32 candies with at least 4 cherry and at least 6 lemon?

Answer 1

Answer:

The number of ways of selecting of n items form r different items =$^{(n+r-1)}C_{(r-1)}$

A store is selling 5 types of hard candies: cherry, strawberry, orange, lemon and pineapple.

A) How many ways are there to choose 32 candies?

So, n = no. of items = 32

r = no. of types of items = 5

So, No. of ways to choose 32 candies = $^{(32+5-1)}C_{(5-1)}$

                                                               = $^{36}C_{4}$

                                                                 = $\frac{36!}{4!(36-4)!}$

                                                                 = $58905$

So, No. of ways to choose 32 candies is 58905

B)32 candies with at least a piece of each flavor?

Out of 32 you choose 5 candies of different types

So, Remaining candies = 32 - 5 = 27

So, No. of ways to choose 27 candies = $^{(27+5-1)}C_{(5-1)}$

                                                               = $^{31}C_{4}$

                                                                 = $\frac{31!}{4!(31-4)!}$

                                                                 = $31465$

So, No. of ways to choose 32 candies with at least a piece of each flavor is 31465

C) 32 candies with at least 4 cherry and at least 6 lemon?

So, you already choose 6+4= 10

So, remaining candies = 32-10 = 22

So, No. of ways to choose 22 candies = $^{(22+5-1)}C_{(5-1)}$

                                                               = $^{26}C_{4}$

                                                                 = $\frac{26!}{4!(26-4)!}$

                                                                 = $14950$

Hence No. of ways to choose 32 candies with at least 4 cherry and at least 6 lemon is 14950