Remember that we can write n as a sum of n ones. How many plus signs do we use? Question: In how many ways may we write n as a sum of a list of k positive numbers? Such a list is called a composition of n intok parts. For example, there are three different ways can we write n=5 as a sum of a list of k=3 positive numbers: • 1+1+3 = 5 1+3+1 = 5 • 3+1+1 = 5 Notice that we want to count all three sums, even though they all use two "1's" and one "3". HINT: How many ways can you insert two "+" signs into the list 1 1 111?
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.
![Remember that we can write n as a sum of n ones. How many plus signs do we use?
Question: In how many ways may we write n as a sum of a list of k positive numbers?
Such a list is called a composition of n into k parts.
For example, there are three different ways can we write n=5 as a sum of a list of k=3 positive
numbers:
• 1+1+3 = 5
1+3+1 = 5
3+1+1 = 5
Notice that we want to count all three sums, even though they all use two "1's" and one "3".
HINT: How many ways can you insert two "+" signs into the list 11111?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2ae01401-271c-489d-8cc3-f6ca5853bc4d%2F42e4f371-705e-4f62-b235-16bcc075d0ee%2Flf5sze4_processed.png&w=3840&q=75)
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