There are 5 different flavors of donuts. A person may choose either 0 or 1 of each flavor (cannot have more than 1 of a flavor). Person A must choose exactly 2 donuts out of the 5 (can't have 2 of the same donut because of the previously mentioned rule). Using combinations, we see that 5C2 = 10 different possible combinations. How would you show the work for the math for this without using combinations and without having to list out all the different personalities? Trying to understand the simple math (addition or multiplication rules?) Similarly, how would I show the math if other people may choose any combination of the 5 flavors, of how many different possibilities are there? 0 donuts = 1 possibility 1 donut = 5C1 = 5 possibilities 2 donuts = 5C2 = 10 possibilities 3 donuts = 5C3 = 10 possibilities 4 donuts = 5C4 = 5 possibilities 5 donuts = 5C5 = 1 possibility 1 + 5 + 10 + 10 + 5 + 1 = 32 possibilities (how would I do the math for above without using combinations?)
Permutations and Combinations
If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.
Counting Theory
The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.
There are 5 different flavors of donuts.
A person may choose either 0 or 1 of each flavor (cannot have more than 1 of a flavor).
Person A must choose exactly 2 donuts out of the 5 (can't have 2 of the same donut because of the previously mentioned rule). Using combinations, we see that 5C2 = 10 different possible combinations. How would you show the work for the math for this without using combinations and without having to list out all the different personalities? Trying to understand the simple math (addition or multiplication rules?)
Similarly, how would I show the math if other people may choose any combination of the 5 flavors, of how many different possibilities are there?
0 donuts = 1 possibility
1 donut = 5C1 = 5 possibilities
2 donuts = 5C2 = 10 possibilities
3 donuts = 5C3 = 10 possibilities
4 donuts = 5C4 = 5 possibilities
5 donuts = 5C5 = 1 possibility
1 + 5 + 10 + 10 + 5 + 1 = 32 possibilities (how would I do the math for above without using combinations?)
Then wanted to double check, if I have to find how many total combinations are there for Person A and "other" people if there are "x" total number of people (i.e. x = 1 means just person A, x = 2 means person A and one "other" person:
10 * 32^(x-1)
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