Question

The "eating club" is hosting a make-your-own sundae at which the following are provided:
Flavors: Chocolate, Cookie, Strawberry, Vanilla
Toppings: Caramel, Hot fudge, Marshmallow, M&M, Nuts, Strawberries.
a) How many sundaes are possible using one flavor and 3 different toppings?
b) How many sundaes are possible using one flavor and from 0 to 6 toppings? (I don't get this part)
c) How many different combinations of flavors of three scoops of flavors are possible if it is permissible to make all 3 scoops the same flavors?
Please, explain.

Answer 1

for part a) I got 4*6C3= 4*20 =80, is it right?

Answer 2

@kirbykirby

Answer 3

part a is right

Answer 4

4C1*6C3

Answer 5

Why do we have to use 4C1 ? is it not that "using 1 flavor" exactly 4 ways?

Answer 6

4C1 : number of ways of choosing 1 flavor from the available 4 flavors

Answer 7

4C1 = 4

Answer 8

4C1 tells explicitly that you're picking 1 flavor from available 4 flavors

4 doesn't tell you anything

Answer 9

got you. Explain me part b, please

Answer 10

you can choose one flavor in : 4C1 ways

Answer 11

next, for toppings, you're free to have any number you want between 0->6

Answer 12

you can have either 0 toppings
or 1 toppings
or 2 toppings
...
or 6 toppings

right ?

Answer 13

Yes

Answer 14

lets assume you're taking all different toppings

Answer 15

then, number of ways of choosing 0 toppings from available 6 toppings = 6C0

Answer 16

number of ways of choosing 1 toppings from available 6 toppings = 6C1

Answer 17

number of ways of choosing 2 toppings from available 6 toppings = 6C2

Answer 18

...

Answer 19

number of ways of choosing 6 toppings from available 6 toppings = 6C6

Answer 20

So total number of ways of choosing 0->6 different toppings = 6C0 + 6C1 + ... + 6C6

Answer 21

adding them all should give you 2^6 = 64

Answer 22

Overall, part b answer would be :

4C1 * 64

Answer 23

Gooooooooooooooot it. :)

Answer 24

Note that we have assumed you're choosing DIFFERENT toppings.

Answer 25

you're not allowed to pick more than 1 of any of the same toppings

Answer 26

for example,
you're not allowed to pick 2 Caramel toppings even if you like Caramel so much

Answer 27

Yes,

Answer 28

Part c is just about the flavor? so that we ignore the toppings part, right?

Answer 29

And although they say "the combination" but "it is permissible to make all 3 scoops the same flavor" means permutation, right?

Answer 30

not exactly

Answer 31

:(

Answer 32

because order doesn't matter when you eat

Answer 33

`Chocolate, Cookie, Strawberry`

is same as

`Cookie, Strawberry, Chocolate`

Answer 34

Yes,

Answer 35

I have to go. Please leave your guidance here. I'll be back to take. Thanks in advance. :)

Answer 36

familiar with stars and bars ?