In a class of 16 students, there are 6 male students and 10 female students. In how many ways can you select a group of 5 students from this class such that 3 of the 5 students are males and the other two are females?
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.
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МАT 255
Note: For Problems 3 and 4, you may use the following two facts about the Euler Phi-function:
a) If integers m and n are coprime, then o(mn) = ¢(m)o(n).
b) If p is a prime number and a is a positive integer, then o(p") = pª-1(p – 1)
1. In a class of 16 students, there are 6 male students and 10 female students. In how many ways can
you select a group of 5 students from this class such that 3 of the 5 students are males and the other
two are females?
2. Use the inclusion-exclusion principle to find the number of solutions to the equation
x + y + z = 19,
where x, y, and z are positive integers. [Hint: You know how to find the number of non-negative
solutions to the equation above.]
3. Let ø: {1, 2, 3, ...} → {1, 2, 3, ...} denote the Euler phi-function. If ø injective? Is it surjective? Justify
your answers.
4. Define a relation - on the set N of natural numbers as follows: For natural numbers n and m, n ~ m if
and only if ø(n) = ¢(m), where o denotes the Euler phi-function. Is ~ defined as such an equivalence
relation on N? If so, find the equivalence class containing 5.
5. Let S be a set containing 8 elements. What is the number of subsets of S containing at least two
elements? Justify your answer.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0b1412fc-5e24-47c1-805c-f79f0dfd1ed2%2Ffdb44ff8-c929-4c60-af7d-43046e7bf5a2%2Fso27neg_processed.jpeg&w=3840&q=75)
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