1. ## Ice cream probability.

Ice cream flavor: Chocolate, Strawberry, Peach, Vanilla

Toppings: Caramel, fudge, Marshmallow, M&Ms, Nuts, Sprinkles

a) How many sundaes are possible using 1 flavor of ice cream and 3 toppings?

b) How many sundaes are possible using 1 flavor of ice cream and from 0 to 6 toppings?

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2. Hi amor_vincit_omnia,

a) Well assuming each flavour can go with each topping then we have to choose one flavour out of 4 and then choose 3 toppings out of 6. The order of the toppings do not matter and thus we can use combinations on the toppings. We are only interested in choosing one flavour and thus we can immediately say there are 4 ways of doing this without the need for cominations. Thus,

$4 {6\choose3}~=~80$

b) Now for each ice cream we can have either 0, 1, 3 etc. toppings. Well firstly just concentrate on the numbers of ways for one ice cream;

$0~Toppings \implies~1~way$
$1~Topping \implies~ 6~ways$
$2~Toppings \implies~ {6\choose2}~ways$
$3~Toppings \implies~ {6\choose3}~ways$
$4~Toppings \implies~ {6\choose4}~ways$
$5~Toppings \implies~ 6~ways$
$6~Toppings \implies~1~way$

Thus the total numbers of ways for one ice cream is;

$1+6+{6\choose2}+{6\choose3}+{6\choose4}+6+1~=~64$

We have 4 ice creams so the total number of ways is $4~\times~64~=~256$

3. Hello, amor_vincit_omnia!

Ice cream flavors: Chocolate, Strawberry, Peach, Vanilla

Toppings: Caramel, fudge, Marshmallow, M&Ms, Nuts, Sprinkles

a) How many sundaes are possible using 1 flavor of ice cream and 3 toppings?
There are ${\color{blue}4}$ choices of ice cream flavors.
Select three toppings from the available six: . ${6\choose3} \:=\:{\color{blue}20}$ ways.

Therefore, there are: . $4\cdot20 \:=\:\boxed{80}$ possible sundaes.

b) How many sundaes are possible using 1 flavor of ice cream and from 0 to 6 toppings?
There are ${\color{blue}4}$ choices of ice cream flavors.

For each of the six toppings, there are two choices: .(1) use it, or (2) don't use it.
. . Hence, there are: . $2^6 \:=\:{\color{blue}64}$ choices.

Therefore, there are: . $4\cdot64 \:=\:\boxed{256}$ possible sundaes.