Question

Need help understanding factorials and their role in probability:

Answer 1

A burger joint offers 3 types of meats and 7 condiments. A burger just include meat, but my include as many or as few condiments as the customer wants. How many different burgers are possible?

A) 8!
B) 3 7!
C) 3 8!
D) 8 2^7
E) 3 2^7

Answer 2

So I know the answer is E but I picked B. I thought it would be 3* 8! because you had 8 choices of condiments (because some may want all or none).

What is the difference in ! and 2^7?

Answer 3

I understand ! means like 3*2*1

Answer 4

let me think about this.
I do know we see factorials when we start with
N objects, and pick them in order.
for example if we have 7 spices, and we choose the first one
7 choices for the first choice.
6 choices for the 2nd
5 for the 3rd
etc.

so factorial implies order. eg. pepper, salt is a different permutation than salt,pepper

Answer 5

Ok. I did not know that. So in this case, since it is a hamburger, there is no "order." They can pick whatever they want however. So we use 2^7. Ok. But what if they did not want condiments? Isn't 0^7 0 and 0*3=0?

Answer 6

But I guess 1^7 is 1 and 3*1 is still 3

Answer 7

the idea of choosing (or not choosing) 7 things is a bit tricky
(and was not obvious at all how to do it)

let's use 2 things (instead of 7): we can pick A,B or A or B or neither
if we use 0 for not pick, and 1 for pick
and assume A is first and B is second, we can write this table
A B
0 0 we picked neither
0 1 we picked B
1 0 we picked A
1 1 we picked both

notice that we can "classify" those 2 choices for A (i.e. 0 or 1) times 2 choices for B
= 4 choices all together
if we had 3 things, the number of choices would be 2*2*2 = 8
or in general 2^n

Answer 8

That makes so much sense. Thanks so much! I'm gonna work more problems like this to solidify it. Thanks again!